View source for Fermat's Last Theorem

{{Short description|17th-century conjecture proved by Andrew Wiles in 1994}}
{{For-multi|other theorems named after Pierre de Fermat|Fermat's theorem|the book by Simon Singh|Fermat's Last Theorem (book)}}
{{External links|date=June 2021}}
{{Use dmy dates|date=August 2023}}
{{Infobox mathematical statement
| name = Fermat's Last Theorem
| image = Diophantus-II-8-Fermat.jpg
| caption = The 1670 edition of [[Diophantus]]'s ''[[Arithmetica]]'' includes Fermat's commentary, referred to as his "Last Theorem" (''Observatio Domini Petri de Fermat''), posthumously published by his son.
| field = [[Number theory]]
| statement = For any integer {{math|''n'' > 2}}, the equation {{math|1=''a''<sup>''n''</sup> + ''b''<sup>''n''</sup> = ''c''<sup>''n''</sup>}} has no positive integer solutions.
| first stated by = [[Pierre de Fermat]]
| first stated date = {{c.}} 1637
| first proof by = [[Andrew Wiles]]
| first proof date = Released 1994<br/>Published 1995
| implied by = {{ubl | [[abc conjecture|Effective abc conjecture]] | [[Szpiro's conjecture#Modified Szpiro conjecture|Effective modified Szpiro conjecture]] | [[Modularity theorem]] }}
| generalizations = {{ubl| [[Beal conjecture]] | [[Fermat–Catalan conjecture]] }}
}}

In [[number theory]], '''Fermat's Last Theorem''' (sometimes called '''Fermat's conjecture''', especially in older texts) states that no three [[positive number|positive]] [[integer]]s {{math|''a''}}, {{math|''b''}}, and {{math|''c''}} satisfy the equation {{math|1=''a''<sup>''n''</sup> + ''b''<sup>''n''</sup> = ''c''<sup>''n''</sup>}} for any integer value of {{math|''n''}} greater than {{math|2}}. The cases {{math|1=''n'' = 1}} and {{math|1=''n'' = 2}} have been known since antiquity to have infinitely many solutions.<ref name="auto">Singh, pp. 18–20</ref>

The proposition was first stated as a theorem by [[Pierre de Fermat]] around 1637 in the margin of a copy of ''[[Arithmetica]]''. Fermat added that he had a proof that was too large to fit in the margin. Although other statements claimed by Fermat without proof were subsequently proven by others and credited as theorems of Fermat (for example, [[Fermat's theorem on sums of two squares]]), Fermat's Last Theorem resisted proof, leading to doubt that Fermat ever had a correct proof. Consequently, the proposition became known as a [[conjecture]] rather than a theorem. After 358 years of effort by mathematicians, [[Wiles's proof of Fermat's Last Theorem|the first successful proof]] was released in 1994 by [[Andrew Wiles]] and formally published in 1995. It was described as a "stunning advance" in the citation for Wiles's [[Abel Prize]] award in 2016.<ref name="abelcitation">{{Cite web |url=http://www.abelprize.no/c67107/binfil/download.php?tid=67059 |title=Abel prize 2016 – full citation |access-date=16 March 2016 |archive-date=20 May 2020 |archive-url=https://web.archive.org/web/20200520195920/https://www.abelprize.no/c67107/binfil/download.php?tid=67059 |url-status=dead }}</ref> It also proved much of the Taniyama–Shimura conjecture, subsequently known as the [[modularity theorem]], and opened up entire new approaches to numerous other problems and mathematically powerful [[lift (mathematics)|modularity lifting]] techniques.

The unsolved problem stimulated the development of [[algebraic number theory]] in the 19th and 20th centuries. It is among the most notable theorems in the [[history of mathematics]] and prior to its proof was in the ''[[Guinness Book of World Records]]'' as the "most difficult mathematical problem", in part because the theorem has the largest number of unsuccessful proofs.<ref>{{cite book|title=The Guinness Book of World Records|url=https://archive.org/details/guinnessbookofwo00mark|url-access=registration|year=1995|section=Science and Technology|publisher=Guinness Publishing Ltd.|isbn=9780965238304}}</ref>

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== Overview ==
=== Pythagorean origins ===
The [[Pythagorean theorem|Pythagorean equation]], {{nowrap|1=''x''<sup>2</sup> + ''y''<sup>2</sup> = ''z''<sup>2</sup>}}, has an infinite number of positive [[integer]] solutions for ''x'', ''y'', and ''z''; these solutions are known as [[Pythagorean triple]]s (with the simplest example being 3, 4, 5). Around 1637, Fermat wrote in the margin of a book that the more general equation {{nowrap|1=''a''<sup>''n''</sup> + ''b''<sup>''n''</sup> = ''c''<sup>''n''</sup>}} had no solutions in positive integers if ''n'' is an integer greater than 2. Although he claimed to have a general [[mathematical proof|proof]] of his conjecture, Fermat left no details of his proof, and none has ever been found. His claim was discovered some 30&nbsp;years later, after his death. This claim, which came to be known as ''Fermat's Last Theorem'', stood unsolved for the next three and a half centuries.<ref name="BostonFLTproof">{{Cite web|url=https://www.math.wisc.edu/~boston/869.pdf |author=Nigel Boston |page=5|title=The Proof of Fermat's Last Theorem}}</ref>

The claim eventually became one of the most notable unsolved problems of mathematics. Attempts to prove it prompted substantial development in [[number theory]], and over time Fermat's Last Theorem gained prominence as an [[List of unsolved problems in mathematics|unsolved problem in mathematics]].

=== Subsequent developments and solution ===
{{more citations needed|section|date=August 2020}}
The special case {{math|1=''n'' = 4}}, proved by Fermat himself, is sufficient to establish that if the theorem is false for some [[exponent]] ''n'' that is not a [[prime number]], it must also be false for some smaller ''n'', so only prime values of ''n'' need further investigation.<ref group=note>If the exponent ''n'' were not prime or 4, then it would be possible to write ''n'' either as a product of two smaller integers ({{nowrap|1=''n'' = ''PQ''}}), in which ''P'' is a prime number greater than 2, and then {{nowrap|1=''a<sup>n</sup>'' = ''a<sup>PQ</sup>'' = (''a<sup>Q</sup>'')<sup>''P''</sup>}} for each of ''a'', ''b'', and ''c''. That is, an equivalent solution would ''also'' have to exist for the prime power ''P'' that is ''smaller'' than ''n''; or else as ''n'' would be a power of 2 greater than 4, and writing {{nowrap|1=''n'' = 4''Q''}}, the same argument would hold.</ref> Over the next two centuries (1637–1839), the conjecture was proved for only the primes 3, 5, and 7, although [[Sophie Germain]] innovated and proved an approach that was relevant to an entire class of primes. In the mid-19th century, [[Ernst Kummer]] extended this and proved the theorem for all [[regular prime]]s, leaving irregular primes to be analyzed individually. Building on Kummer's work and using sophisticated computer studies, other mathematicians were able to extend the proof to cover all prime exponents up to four million,<ref name=":0"/> but a proof for all exponents was inaccessible (meaning that mathematicians generally considered a proof impossible, exceedingly difficult, or unachievable with current knowledge).<ref>Singh, p. 223</ref>

Separately, around 1955, Japanese mathematicians [[Goro Shimura]] and [[Yutaka Taniyama]] suspected a link might exist between [[elliptic curve]]s and [[modular form]]s, two completely different areas of mathematics. Known at the time as the [[Taniyama–Shimura conjecture]] (eventually as the modularity theorem), it stood on its own, with no apparent connection to Fermat's Last Theorem. It was widely seen as significant and important in its own right, but was (like Fermat's theorem) widely considered completely inaccessible to proof.<ref>Singh 1997, pp. 203–205, 223, 226</ref>

In 1984, [[Gerhard Frey]] noticed an apparent link between these two previously unrelated and unsolved problems. An outline suggesting this could be proved was given by Frey. The full proof that the two problems were closely linked was accomplished in 1986 by [[Ken Ribet]], building on a partial proof by [[Jean-Pierre Serre]], who proved all but one part known as the "epsilon conjecture" (see: ''[[Ribet's Theorem]]'' and ''[[Frey curve]]'').<ref name="abelcitation"/> These papers by Frey, Serre and Ribet showed that if the Taniyama–Shimura conjecture could be proven for at least the semi-stable class of elliptic curves, a proof of Fermat's Last Theorem would also follow automatically. The connection is described [[#Equivalent statements of the theorem|below]]: any solution that could contradict Fermat's Last Theorem could also be used to contradict the Taniyama–Shimura conjecture. So if the modularity theorem were found to be true, then by definition, no solution contradicting Fermat's Last Theorem could exist, meaning that Fermat's Last Theorem must also be true.

Although both problems were daunting and widely considered to be "completely inaccessible" to proof at the time,<ref name="abelcitation"/> this was the first suggestion of a route by which Fermat's Last Theorem could be extended and proved for all numbers, not just some numbers. Unlike Fermat's Last Theorem, the Taniyama–Shimura conjecture was a major active research area and viewed as more within reach of contemporary mathematics.<ref>Singh, p. 144 quotes Wiles's reaction to this news: "I was electrified. I knew that moment that the course of my life was changing because this meant that to prove Fermat's Last Theorem all I had to do was to prove the Taniyama–Shimura conjecture. It meant that my childhood dream was now a respectable thing to work on."</ref> However, general opinion was that this simply showed the impracticality of proving the Taniyama–Shimura conjecture.<ref name="singh144">Singh, p. 144</ref> Mathematician [[John H. Coates|John Coates]]' quoted reaction was a common one:<ref name="singh144"/>

{{blockquote|I myself was very sceptical that the beautiful link between Fermat's Last Theorem and the Taniyama–Shimura conjecture would actually lead to anything, because I must confess I did not think that the Taniyama–Shimura conjecture was accessible to proof. Beautiful though this problem was, it seemed impossible to actually prove. I must confess I thought I probably wouldn't see it proved in my lifetime.}}

On hearing that Ribet had proven Frey's link to be correct, English mathematician [[Andrew Wiles]], who had a childhood fascination with Fermat's Last Theorem and had a background of working with elliptic curves and related fields, decided to try to prove the Taniyama–Shimura conjecture as a way to prove Fermat's Last Theorem. In 1993, after six years of working secretly on the problem, Wiles [[Wiles's proof of Fermat's Last Theorem|succeeded in proving]] enough of the conjecture to prove Fermat's Last Theorem. Wiles's paper was massive in size and scope. A flaw was discovered in one part of his original paper during [[peer review]] and required a further year and collaboration with a past student, [[Richard Taylor (mathematician)|Richard Taylor]], to resolve. As a result, the final proof in 1995 was accompanied by a smaller joint paper showing that the fixed steps were valid. Wiles's achievement was reported widely in the popular press, and was popularized in books and television programs. The remaining parts of the Taniyama–Shimura–Weil conjecture, now proven and known as the modularity theorem, were subsequently proved by other mathematicians, who built on Wiles's work between 1996 and 2001.<ref name="Diamond-1996">{{Cite journal|last=Diamond|first=Fred|date=July 1996|title=On Deformation Rings and Hecke Rings|url=https://www.jstor.org/stable/2118586|journal=The Annals of Mathematics|volume=144|issue=1|pages=137–166|doi=10.2307/2118586|jstor=2118586}}</ref><ref name="Conrad-etal-1999">{{Cite journal|last1=Conrad|first1=Brian|last2=Diamond|first2=Fred|last3=Taylor|first3=Richard|date=1999|title=Modularity of certain potentially Barsotti-Tate Galois representations|url=https://www.ams.org/jams/1999-12-02/S0894-0347-99-00287-8/|journal=Journal of the American Mathematical Society|language=en|volume=12|issue=2|pages=521–567|doi=10.1090/S0894-0347-99-00287-8|issn=0894-0347|doi-access=free}}</ref><ref name="Breuil-etal-2001">{{Cite journal|last1=Breuil|first1=Christophe|last2=Conrad|first2=Brian|last3=Diamond|first3=Fred|last4=Taylor|first4=Richard|date=2001-05-15|title=On the modularity of elliptic curves over '''Q''': Wild 3-adic exercises|url=https://www.ams.org/jams/2001-14-04/S0894-0347-01-00370-8/|journal=Journal of the American Mathematical Society|language=en|volume=14|issue=4|pages=843–939|doi=10.1090/S0894-0347-01-00370-8|issn=0894-0347|doi-access=free}}</ref> For his proof, Wiles was honoured and received numerous awards, including the 2016 [[Abel Prize]].<ref name="natureAbel">{{cite journal |title=Fermat's last theorem earns Andrew Wiles the Abel Prize |journal=[[Nature (journal)|Nature]] |volume=531 |issue=7594 |pages=287 |date=15 March 2016 |doi=10.1038/nature.2016.19552 |pmid=26983518 |last=Castelvecchi |first=Davide |bibcode=2016Natur.531..287C |s2cid=4383161 |doi-access=free }}</ref><ref>[https://web.archive.org/web/20160315135239/https://www.washingtonpost.com/world/europe/british-mathematician-sir-andrew-wiles-gets-abel-math-prize/2016/03/15/41146a7e-eaa9-11e5-a9ce-681055c7a05f_story.html British mathematician Sir Andrew Wiles gets Abel math prize] – The Washington Post.</ref><ref>[http://www.cnn.com/2016/03/16/europe/fermats-last-theorem-solved-math-abel-prize/index.html 300-year-old math question solved, professor wins $700k] – CNN.com.</ref>

=== Equivalent statements of the theorem ===
There are several alternative ways to state Fermat's Last Theorem that are mathematically equivalent to the original statement of the problem.

In order to state them, we use the following notations: let {{math|'''N'''}} be the set of natural numbers 1, 2, 3, ..., let {{math|'''Z'''}} be the set of integers 0, ±1, ±2, ..., and let {{math|'''Q'''}} be the set of rational numbers {{math|''a''/''b''}}, where {{mvar|a}} and {{mvar|b}} are in {{math|'''Z'''}} with {{math|''b'' ≠ 0}}. In what follows we will call a solution to {{math|''x''<sup>''n''</sup> + ''y''<sup>''n''</sup> {{=}} ''z''<sup>''n''</sup>}} where one or more of {{mvar|x}}, {{mvar|y}}, or {{mvar|z}} is zero a ''trivial solution''. A solution where all three are nonzero will be called a ''non-trivial'' solution.

For comparison's sake we start with the original formulation.
* '''Original statement'''. With {{mvar|n}}, {{mvar|x}}, {{mvar|y}}, {{mvar|z}} ∈ {{math|'''N'''}} (meaning that ''n'', ''x'', ''y'', ''z'' are all positive whole numbers) and {{math|''n'' > 2}}, the equation {{math|''x''<sup>''n''</sup> + ''y''<sup>''n''</sup> {{=}} ''z''<sup>''n''</sup>}} has no solutions.

Most popular treatments of the subject state it this way. It is also commonly stated over {{math|'''Z'''}}:<ref>{{cite web |last=Weisstein |first=Eric W. |title=Fermat's Last Theorem. |url=https://mathworld.wolfram.com/FermatsLastTheorem.html |website=MathWorld – A Wolfram Web Resource |access-date=7 May 2021}}</ref>
* '''Equivalent statement 1:''' {{math|''x''<sup>''n''</sup> + ''y''<sup>''n''</sup> {{=}} ''z''<sup>''n''</sup>}}, where integer {{mvar|n}} ≥ 3, has no non-trivial solutions {{mvar|x}}, {{mvar|y}}, {{mvar|z}} ∈ {{math|'''Z'''}}.

The equivalence is clear if {{mvar|n}} is even. If {{mvar|n}} is odd and all three of {{math|''x'', ''y'', ''z''}} are negative, then we can replace {{math|''x'', ''y'', ''z''}} with {{math|−''x'', −''y'', −''z''}} to obtain a solution in {{math|'''N'''}}. If two of them are negative, it must be {{mvar|x}} and {{mvar|z}} or {{mvar|y}} and {{mvar|z}}. If {{math|''x'', ''z''}} are negative and {{mvar|y}} is positive, then we can rearrange to get {{math|(−''z'')<sup>''n''</sup> + ''y''<sup>''n''</sup> {{=}} (−''x'')<sup>''n''</sup>}} resulting in a solution in {{math|'''N'''}}; the other case is dealt with analogously. Now if just one is negative, it must be {{mvar|x}} or {{mvar|y}}. If {{mvar|x}} is negative, and {{mvar|y}} and {{mvar|z}} are positive, then it can be rearranged to get {{math|(−''x'')<sup>''n''</sup> + ''z''<sup>''n''</sup> {{=}} ''y''<sup>''n''</sup>}} again resulting in a solution in {{math|'''N'''}}; if {{mvar|y}} is negative, the result follows symmetrically. Thus in all cases a nontrivial solution in {{math|'''Z'''}} would also mean a solution exists in {{math|'''N'''}}, the original formulation of the problem.
* '''Equivalent statement 2:''' {{math|''x''<sup>''n''</sup> + ''y''<sup>''n''</sup> {{=}} ''z''<sup>''n''</sup>}}, where integer {{math|''n'' ≥ 3}}, has no non-trivial solutions {{math|''x'', ''y'', ''z'' ∈ '''Q'''}}.

This is because the exponents of {{math|''x'', ''y'',}} and {{mvar|z}} are equal (to {{mvar|n}}), so if there is a solution in {{math|'''Q'''}}, then it can be multiplied through by an appropriate common denominator to get a solution in {{math|'''Z'''}}, and hence in {{math|'''N'''}}.
* '''Equivalent statement 3:''' {{math|''x''<sup>''n''</sup> + ''y''<sup>''n''</sup> {{=}} 1}}, where integer {{math|''n'' ≥ 3}}, has no non-trivial solutions {{math|''x'', ''y'' ∈ '''Q'''}}.

A non-trivial solution {{mvar|a}}, {{mvar|b}}, {{mvar|c}} ∈ {{math|'''Z'''}} to {{math|''x''<sup>''n''</sup> + ''y''<sup>''n''</sup> {{=}} ''z''<sup>''n''</sup>}} yields the non-trivial solution {{math|''a''/''c''}}, {{math|''b''/''c''}} ∈ {{math|'''Q'''}} for {{math|''v''<sup>''n''</sup> + ''w''<sup>''n''</sup> {{=}} 1}}. Conversely, a solution {{math|''a''/''b''}}, {{math|''c''/''d''}} ∈ {{math|'''Q'''}} to {{math|''v''<sup>''n''</sup> + ''w''<sup>''n''</sup> {{=}} 1}} yields the non-trivial solution {{math|''ad'', ''cb'', ''bd''}} for {{math|''x''<sup>''n''</sup> + ''y''<sup>''n''</sup> {{=}} ''z''<sup>''n''</sup>}}.

This last formulation is particularly fruitful, because it reduces the problem from a problem about surfaces in three dimensions to a problem about curves in two dimensions. Furthermore, it allows working over the field {{math|'''Q'''}}, rather than over the ring {{math|'''Z'''}}; [[Field (mathematics)|fields]] exhibit more structure than [[Ring (mathematics)|rings]], which allows for deeper analysis of their elements.
* '''Equivalent statement 4 – connection to elliptic curves:''' If {{mvar|a}}, {{mvar|b}}, {{mvar|c}} is a non-trivial solution to {{math|''a''<sup>''p''</sup> + ''b''<sup>''p''</sup> {{=}} ''c''<sup>''p''</sup>}}, {{mvar|p}} odd prime, then {{math|''y''<sup>''2''</sup> {{=}} ''x''(''x'' − ''a''<sup>''p''</sup>)(''x'' + ''b''<sup>''p''</sup>)}} ([[Frey curve]]) will be an [[elliptic curve]] without a modular form.<ref>{{cite journal |last=Wiles |first=Andrew |authorlink=Andrew Wiles |year=1995 |title=Modular elliptic curves and Fermat's Last Theorem |url=http://math.stanford.edu/~lekheng/flt/wiles.pdf |quotation=Frey's suggestion, in the notation of the following theorem, was to show that the (hypothetical) elliptic curve {{math|''y''<sup>''2''</sup> {{=}} ''x''(''x'' + ''u''<sup>''p''</sup>)(''x'' – ''v''<sup>''p''</sup>)}} could not be modular. |journal=[[Annals of Mathematics]] |volume=141 |issue=3 |page=448 |oclc=37032255 |doi=10.2307/2118559 |jstor=2118559 |access-date=11 August 2003 |archive-date=10 May 2011 |archive-url=https://web.archive.org/web/20110510062158/http://math.stanford.edu/%7Elekheng/flt/wiles.pdf |url-status=dead }}</ref>

Examining this elliptic curve with [[Ribet's theorem]] shows that it does not have a [[modular form]]. However, the proof by Andrew Wiles proves that any equation of the form {{math|''y''<sup>''2''</sup> {{=}} ''x''(''x'' − ''a''<sup>''n''</sup>)(''x'' + ''b''<sup>''n''</sup>)}} does have a modular form. Any non-trivial solution to {{math|{{itco|''x''}}<sup>''p''</sup> + {{itco|''y''}}<sup>''p''</sup> {{=}} {{itco|''z''}}<sup>''p''</sup>}} (with {{mvar|p}} an odd prime) would therefore create a [[proof by contradiction|contradiction]], which in turn proves that no non-trivial solutions exist.<ref>{{cite journal |last=Ribet |first=Ken |authorlink=Ken Ribet |title=On modular representations of Gal({{overline|'''Q'''}}/'''Q''') arising from modular forms |journal=Inventiones Mathematicae |volume=100 |year=1990 |issue=2 |page=432 |doi=10.1007/BF01231195 |mr=1047143 |url=http://math.berkeley.edu/~ribet/Articles/invent_100.pdf |bibcode=1990InMat.100..431R|hdl=10338.dmlcz/147454 |s2cid=120614740 }}</ref>

In other words, any solution that could contradict Fermat's Last Theorem could also be used to contradict the modularity theorem. So if the modularity theorem were found to be true, then it would follow that no contradiction to Fermat's Last Theorem could exist either. As described above, the discovery of this equivalent statement was crucial to the eventual solution of Fermat's Last Theorem, as it provided a means by which it could be "attacked" for all numbers at once.

== Mathematical history ==
=== Pythagoras and Diophantus ===
==== Pythagorean triples ====
{{Main|Pythagorean triple}}

In ancient times it was known that a triangle whose sides were in the ratio 3:4:5 would have a right angle as one of its angles. This was used in construction and later in early geometry. It was also known to be one example of a general rule that any triangle where the length of two sides, each squared and then added together {{nowrap|1=(3<sup>2</sup> + 4<sup>2</sup> = 9 + 16 = 25)}}, equals the square of the length of the third side {{nowrap|1=(5<sup>2</sup> = 25)}}, would also be a right angle triangle. This is now known as the [[Pythagorean theorem]], and a triple of numbers that meets this condition is called a Pythagorean triple; both are named after the ancient Greek [[Pythagoras]]. Examples include (3, 4, 5) and (5, 12, 13). There are infinitely many such triples,<ref name="Stillwell_2003">{{cite book | author = Stillwell J | year = 2003 | title = Elements of Number Theory | url = https://books.google.com/books?id=LiAlZO2ntKAC&pg=PA110 | publisher = Springer-Verlag | location = New York | isbn = 0-387-95587-9 | pages = 110–112 | access-date = 2016-03-17| authorlink=John Stillwell }}</ref> and methods for generating such triples have been studied in many cultures, beginning with the [[Babylonian mathematics|Babylonians]]{{sfn|Aczel|1996|pp=13–15|ps=}} and later [[Greek mathematics|ancient Greek]], [[Chinese mathematics|Chinese]], and [[Indian mathematics|Indian]] mathematicians.<ref name="auto"/> Mathematically, the definition of a Pythagorean triple is a set of three integers {{nowrap|(''a'', ''b'', ''c'')}} that satisfy the equation{{sfn|Stark|1978|pp=151–155|ps=}} {{nowrap|1=''a''<sup>2</sup> + ''b''<sup>2</sup> = ''c''<sup>2</sup>}}.

==== Diophantine equations ====
{{Main|Diophantine equation}}

Fermat's equation, {{nowrap|1=''x''<sup>''n''</sup> + ''y''<sup>''n''</sup> = ''z''<sup>''n''</sup>}} with positive [[integer]] solutions, is an example of a [[Diophantine equation]],{{sfn|Stark|1978|pp=145–146|ps=}} named for the 3rd-century [[Alexandria]]n mathematician, [[Diophantus]], who studied them and developed methods for the solution of some kinds of Diophantine equations. A typical Diophantine problem is to find two integers ''x'' and ''y'' such that their sum, and the sum of their squares, equal two given numbers ''A'' and ''B'', respectively:
: <math>A = x + y</math>
: <math>B = x^2 + y^2.</math>

Diophantus's major work is the ''[[Arithmetica]]'', of which only a portion has survived.<ref>Singh, pp. 50–51</ref> Fermat's conjecture of his Last Theorem was inspired while reading a new edition of the ''Arithmetica'',{{sfn|Stark|1978|p=145|ps=}} that was translated into Latin and published in 1621 by [[Claude Gaspard Bachet de Méziriac|Claude Bachet]].{{sfn|Aczel|1996|pp=44–45|ps=}}<ref>Singh, pp. 56–58</ref>

Diophantine equations have been studied for thousands of years. For example, the solutions to the quadratic Diophantine equation {{nowrap|1=''x''<sup>2</sup> + ''y''<sup>2</sup> = ''z''<sup>2</sup>}} are given by the [[Pythagorean triple]]s, originally solved by the Babylonians ({{circa|1800 BC}}).{{sfn|Aczel|1996|pp=14–15|ps=}} Solutions to linear Diophantine equations, such as {{nowrap|1=26''x'' + 65''y'' = 13}}, may be found using the [[Euclidean algorithm]] (c. 5th century BC).{{sfn|Stark|1978|pp=44–47|ps=}}
Many [[Diophantine equation]]s have a form similar to the equation of Fermat's Last Theorem from the point of view of algebra, in that they have no ''cross terms'' mixing two letters, without sharing its particular properties. For example, it is known that there are infinitely many positive integers ''x'', ''y'', and ''z'' such that {{nowrap|1=''x''<sup>''n''</sup> + ''y''<sup>''n''</sup> = ''z''<sup>''m''</sup>}}, where ''n'' and ''m'' are [[relatively prime]] natural numbers.<ref group="note">For example, {{nowrap|((''j''<sup>''r''</sup> + 1)<sup>''s''</sup>)<sup>''r''</sup> + (''j''(''j''<sup>''r''</sup> + 1)<sup>''s''</sup>)<sup>''r''</sup>}} = {{nowrap|(''j''<sup>''r''</sup> + 1)<sup>''rs''+1</sup>}}.</ref>

=== Fermat's conjecture ===
[[File:Diophantus-II-8.jpg|thumb|Problem II.8 in the 1621 edition of the {{lang|la|Arithmetica}} of [[Diophantus]]. On the right is the margin that was too small to contain Fermat's alleged proof of his "last theorem".]]

Problem II.8 of the {{lang|la|[[Arithmetica]]}} asks how a given square number is split into two other squares; in other words, for a given [[rational number]] ''k'', find rational numbers ''u'' and ''v'' such that {{nowrap|1=''k''<sup>2</sup> = ''u''<sup>2</sup> + ''v''<sup>2</sup>}}. Diophantus shows how to solve this sum-of-squares problem for {{nowrap|1=''k'' = 4}} (the solutions being {{nowrap|1=''u'' = 16/5}} and {{nowrap|1=''v'' = 12/5}}).{{sfn|Friberg|2007|pp=333–334|ps=}}

Around 1637, Fermat wrote his Last Theorem in the margin of his copy of the {{lang|la|Arithmetica}} next to [[Diophantus II.VIII|Diophantus's sum-of-squares problem]]:{{sfn|Dickson|1919|p=731|ps=}}<ref>Singh, pp. 60–62</ref>{{sfn|Aczel|1996|p=9|ps=}}

{{Verse translation|lang=la|Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos & generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.|It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second, into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.<ref>T. Heath, ''Diophantus of Alexandria'' Second Edition, Cambridge University Press, 1910, reprinted by Dover, NY, 1964, pp.&nbsp;144–145</ref>{{sfn|Manin|Panchishkin|2007|loc=[https://books.google.com/books?id=wvK586IxaxwC&dq=%22Cubum+autem+in+duos+cubos%2C+aut+quadratoquadratum%22&pg=PA341 p. 341]|ps=}}}}

After Fermat's death in 1665, his son Clément-Samuel Fermat produced a new edition of the book (1670) augmented with his father's comments.<ref>Singh, pp. 62–66</ref> Although not actually a theorem at the time (meaning a mathematical statement for which [[mathematical proof|proof]] exists), the marginal note became known over time as ''Fermat's Last Theorem'',{{sfn|Dickson|1919|p=731|ps=}} as it was the last of Fermat's asserted theorems to remain unproved.<ref>Singh, p. 67</ref>{{sfn|Aczel|1996|p=10|ps=}}

It is not known whether Fermat had actually found a valid proof for all exponents ''n'', but it appears unlikely. Only one related proof by him has survived, namely for the case {{nowrap|1=''n'' = 4}}, as described in the section {{slink||Proofs for specific exponents}}.

While Fermat posed the cases of {{nowrap|1=''n'' = 4}} and of {{nowrap|1=''n'' = 3}} as challenges to his mathematical correspondents, such as [[Marin Mersenne]], [[Blaise Pascal]], and [[John Wallis]],<ref>Ribenboim, pp. 13, 24</ref> he never posed the general case.<ref name="van der Poorten-p.5">van der Poorten, Notes and Remarks 1.2, p. 5</ref> Moreover, in the last thirty years of his life, Fermat never again wrote of his "truly marvelous proof" of the general case, and never published it. Van der Poorten<ref name="van der Poorten-p.5"/> suggests that while the absence of a proof is insignificant, the lack of challenges means Fermat realised he did not have a proof; he quotes [[André Weil|Weil]]<ref>{{cite book | author = André Weil | year = 1984 | title = Number Theory: An approach through history. From Hammurapi to Legendre | publisher = Birkhäuser | location = Basel, Switzerland | page = 104| authorlink=André Weil }}</ref> as saying Fermat must have briefly deluded himself with an irretrievable idea. The techniques Fermat might have used in such a "marvelous proof" are unknown.

Wiles and Taylor's proof relies on 20th-century techniques.<ref>{{cite AV media|url=https://www.youtube.com/watch?v=Ua8K8HW2Qsg#t=47m|title=BBC Documentary}}{{cbignore}}{{Dead YouTube link|date=February 2022}}</ref> Fermat's proof would have had to be elementary by comparison, given the mathematical knowledge of his time.

While [[Harvey Friedman (mathematician)|Harvey Friedman]]'s [[grand conjecture]] implies that any provable theorem (including Fermat's last theorem) can be proved using only '[[elementary function arithmetic]]', such a proof need be 'elementary' only in a technical sense and could involve millions of steps, and thus be far too long to have been Fermat's proof.

=== Proofs for specific exponents ===
{{Main|Proof of Fermat's Last Theorem for specific exponents}}
[[File:Diophantus-VI-24-20-Fermat.png|thumb|right|Fermat's [[infinite descent]] for Fermat's Last Theorem case n=4 in the 1670 edition of the ''Arithmetica'' of [[Diophantus]] (pp. 338–339).]]

==== Exponent = 4 ====
Only one relevant [[Fermat's right triangle theorem|proof by Fermat]] has survived, in which he uses the technique of [[infinite descent]] to show that the area of a right triangle with integer sides can never equal the square of an integer.<ref>{{cite web | author = Freeman L | title = Fermat's One Proof | url = http://fermatslasttheorem.blogspot.com/2005/05/fermats-one-proof.html | access-date = 23 May 2009| date = 12 May 2005 }}</ref>{{sfn|Dickson|1919|pp=615–616|ps=}}{{sfn|Aczel|1996|p=44|ps=}} His proof is equivalent to demonstrating that the equation
: <math>x^4 - y^4 = z^2</math>
has no primitive solutions in integers (no pairwise [[coprime]] solutions). In turn, this proves Fermat's Last Theorem for the case {{nowrap|1=''n'' = 4}}, since the equation {{nowrap|1=''a''<sup>4</sup> + ''b''<sup>4</sup> = ''c''<sup>4</sup>}} can be written as {{nowrap|1=''c''<sup>4</sup> − ''b''<sup>4</sup> = (''a''<sup>2</sup>)<sup>2</sup>}}.

Alternative proofs of the case {{nowrap|1=''n'' = 4}} were developed later<ref>Ribenboim, pp. 15–24</ref> by [[Bernard Frénicle de Bessy|Frénicle de Bessy]] (1676),<ref>Frénicle de Bessy, ''Traité des Triangles Rectangles en Nombres'', vol. I, 1676, Paris. Reprinted in ''Mém. Acad. Roy. Sci.'', '''5''', 1666–1699 (1729)</ref> [[Leonhard Euler]] (1738),<ref name="Euler_1738">{{cite journal | author = Euler L | year = 1738 | title = Theorematum quorundam arithmeticorum demonstrationes | journal = Novi Commentarii Academiae Scientiarum Petropolitanae | volume = 10 | pages = 125–146| authorlink=Leonhard Euler }}. Reprinted ''Opera omnia'', ser. I, "Commentationes Arithmeticae", vol. I, pp. 38–58, Leipzig:Teubner (1915)</ref> Kausler (1802),<ref name="Kausler_1802"/> [[Peter Barlow (mathematician)|Peter Barlow]] (1811),<ref>{{cite book | author = Barlow P | year = 1811 | title = An Elementary Investigation of Theory of Numbers | location = St. Paul's Church-Yard, London | publisher = J. Johnson | pages = 144–145| authorlink=Peter Barlow (mathematician) }}</ref> [[Adrien-Marie Legendre]] (1830),<ref name="Legendre_1830"/> Schopis (1825),<ref>{{cite book | author = Schopis | year = 1825 | title = Einige Sätze aus der unbestimmten Analytik | publisher = Programm | location = Gummbinnen}}</ref> [[Olry Terquem]] (1846),<ref>{{cite journal | author = Terquem O | year = 1846 | title = Théorèmes sur les puissances des nombres | journal = [[Nouvelles Annales de Mathématiques]] | volume = 5 | pages = 70–87| authorlink=Olry Terquem }}</ref> [[Joseph Louis François Bertrand|Joseph Bertrand]] (1851),<ref>{{cite book | author = Bertrand J | year = 1851 | title = Traité Élémentaire d'Algèbre | publisher = Hachette | location = Paris | pages = 217–230, 395| authorlink=Joseph Louis François Bertrand }}</ref> [[Victor Lebesgue]] (1853, 1859, 1862),<ref>{{cite journal | author = Lebesgue VA | year = 1853 | title = Résolution des équations biquadratiques ''z''<sup>2</sup> = ''x''<sup>4</sup> ± 2<sup>''m''</sup>''y''<sup>4</sup>, ''z''<sup>2</sup> = 2<sup>''m''</sup>''x''<sup>4</sup> − ''y''<sup>4</sup>, 2<sup>''m''</sup>''z''<sup>2</sup> = ''x''<sup>4</sup> ± ''y''<sup>4</sup> | journal = [[Journal de Mathématiques Pures et Appliquées]] | volume = 18 | pages = 73–86}}<br/>{{cite book | author = Lebesgue VA | year = 1859 | title = Exercices d'Analyse Numérique | publisher = Leiber et Faraguet | location = Paris | pages = 83–84, 89}}<br/>{{cite book | author = Lebesgue VA | year = 1862 | title = Introduction à la Théorie des Nombres | publisher = Mallet-Bachelier | location = Paris | pages = 71–73}}</ref> [[Théophile Pépin]] (1883),<ref>{{cite journal | author = Pepin T | year = 1883 | title = Étude sur l'équation indéterminée ''ax''<sup>4</sup> + ''by''<sup>4</sup> = ''cz''<sup>2</sup> | journal = Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Serie IX. Matematica e Applicazioni. | volume = 36 | pages = 34–70}}</ref> Tafelmacher (1893),<ref>{{cite journal | author = A. Tafelmacher | year = 1893 | title = Sobre la ecuación ''x''<sup>4</sup> + ''y''<sup>4</sup> = ''z''<sup>4</sup> | journal = [[Anales de la Universidad de Chile]] | volume = 84 | pages = 307–320 |url=https://anales.uchile.cl/index.php/ANUC/article/view/20645}}</ref> [[David Hilbert]] (1897),<ref>{{cite journal | author = Hilbert D | year = 1897 | title = Die Theorie der algebraischen Zahlkörper | journal = [[Jahresbericht der Deutschen Mathematiker-Vereinigung]] | volume = 4 | pages = 175–546| authorlink=David Hilbert }} Reprinted in 1965 in ''Gesammelte Abhandlungen, vol. I'' by New York:Chelsea.</ref> Bendz (1901),<ref>{{cite thesis | author = Bendz TR | year = 1901 | title = Öfver diophantiska ekvationen {{nowrap|1=x<sup>n</sup> + y<sup>n</sup> = z<sup>n</sup>}} | publisher = Almqvist & Wiksells Boktrycken | location = Uppsala}}</ref> Gambioli (1901),<ref name="Gambioli_1901"/> [[Leopold Kronecker]] (1901),<ref>{{cite book | author = Kronecker L | year = 1901 | title = Vorlesungen über Zahlentheorie, vol. I | publisher = Teubner | location = Leipzig | pages = 35–38| authorlink=Leopold Kronecker }} Reprinted by New York:Springer-Verlag in 1978.</ref> Bang (1905),<ref>{{cite journal | author = Bang A | year = 1905 | title = Nyt Bevis for at Ligningen {{nowrap|1=''x''<sup>4</sup> − ''y''<sup>4</sup> = ''z''<sup>4</sup>}}, ikke kan have rationale Løsinger | journal = Nyt Tidsskrift for Matematik | volume = 16B | pages = 31–35 |jstor=24528323}}</ref> Sommer (1907),<ref>{{cite book | author = Sommer J | year = 1907 | title = Vorlesungen über Zahlentheorie | publisher = Teubner | location = Leipzig}}</ref> Bottari (1908),<ref>{{cite journal | author = Bottari A | year = 1908 | title = Soluzione intere dell'equazione pitagorica e applicazione alla dimostrazione di alcune teoremi della teoria dei numeri | journal = Periodico di Matematiche | volume = 23 | pages = 104–110}}</ref> [[Karel Rychlík]] (1910),<ref name="Rychlik_1910"/> Nutzhorn (1912),<ref>{{cite journal | author = Nutzhorn F | year = 1912 | title = Den ubestemte Ligning ''x''<sup>4</sup> + ''y''<sup>4</sup> = ''z''<sup>4</sup> | journal = Nyt Tidsskrift for Matematik | volume = 23B | pages = 33–38}}</ref> [[Robert Daniel Carmichael|Robert Carmichael]] (1913),<ref>{{cite journal | author = Carmichael RD | year = 1913 | title = On the impossibility of certain Diophantine equations and systems of equations | journal = [[American Mathematical Monthly]] | volume = 20 | pages = 213–221 | doi = 10.2307/2974106 | issue = 7 | publisher = Mathematical Association of America | jstor = 2974106| authorlink=Robert Daniel Carmichael }}</ref> Hancock (1931),<ref>{{cite book | author = Hancock H | year = 1931 | title = Foundations of the Theory of Algebraic Numbers, vol. I | publisher = Macmillan | location = New York}}</ref> [[Gheorghe Vrănceanu]] (1966),<ref>{{cite journal | author = Gheorghe Vrănceanu | year = 1966 | title = Asupra teorema lui Fermat pentru {{nowrap|1=''n'' = 4}} | journal = Gazeta Matematică Seria A | volume = 71 | pages = 334–335 }} Reprinted in 1977 in ''Opera matematica'', vol. 4, pp. 202–205, București: Editura Academiei Republicii Socialiste România.</ref> Grant and Perella (1999),<ref>Grant, Mike, and Perella, Malcolm, "Descending to the irrational", ''Mathematical Gazette'' 83, July 1999, pp. 263–267.</ref> Barbara (2007),<ref>Barbara, Roy, "Fermat's last theorem in the case {{nowrap|1=''n'' = 4}}", ''Mathematical Gazette'' 91, July 2007, 260–262.</ref> and Dolan (2011).<ref>Dolan, Stan, "Fermat's method of ''descente infinie''", ''[[Mathematical Gazette]]'' 95, July 2011, 269–271.</ref>

==== Other exponents ====
After Fermat proved the special case {{nowrap|1=''n'' = 4}}, the general proof for all ''n'' required only that the theorem be established for all odd prime exponents.<ref>Ribenboim, pp. 1–2</ref> In other words, it was necessary to prove only that the equation {{nowrap|1=''a''<sup>''n''</sup> + ''b''<sup>''n''</sup> = ''c''<sup>''n''</sup>}} has no positive integer solutions {{nowrap|(''a'', ''b'', ''c'')}} when ''n'' is an odd [[prime number]]. This follows because a solution {{nowrap|(''a'', ''b'', ''c'')}} for a given ''n'' is equivalent to a solution for all the factors of ''n''. For illustration, let ''n'' be factored into ''d'' and ''e'', ''n''&nbsp;=&nbsp;''de''. The general equation
: ''a''<sup>''n''</sup> + ''b''<sup>''n''</sup> = ''c''<sup>''n''</sup>
implies that {{nowrap|(''a''<sup>''d''</sup>, ''b''<sup>''d''</sup>, ''c''<sup>''d''</sup>)}} is a solution for the exponent ''e''
: (''a''<sup>''d''</sup>)<sup>''e''</sup> + (''b''<sup>''d''</sup>)<sup>''e''</sup> = (''c''<sup>''d''</sup>)<sup>''e''</sup>.

Thus, to prove that Fermat's equation has no solutions for {{nowrap|''n'' > 2}}, it would suffice to prove that it has no solutions for at least one prime factor of every ''n''. Each integer {{nowrap|''n'' > 2}} is divisible by 4 or by an odd prime number (or both). Therefore, Fermat's Last Theorem could be proved for all ''n'' if it could be proved for {{nowrap|1=''n'' = 4}} and for all odd primes ''p''.

In the two centuries following its conjecture (1637–1839), Fermat's Last Theorem was proved for three odd prime exponents ''p''&nbsp;=&nbsp;3, 5 and 7. The case {{nowrap|1=''p'' = 3}} was first stated by [[Abu-Mahmud Khojandi]] (10th century), but his attempted proof of the theorem was incorrect.{{sfn|Dickson|1919|p=545|ps=}}<ref>{{MacTutor|id=Al-Khujandi|title=Abu Mahmud Hamid ibn al-Khidr Al-Khujandi|mode=cs1}}</ref> In 1770, [[Leonhard Euler]] gave a proof of ''p''&nbsp;=&nbsp;3,<ref>[[Leonhard Euler|Euler L]] (1770) ''Vollständige Anleitung zur Algebra'', Roy. Acad. Sci., St. Petersburg.</ref> but his proof by infinite descent<ref>{{cite web | author = Freeman L | title = Fermat's Last Theorem: Proof for ''n''&nbsp;=&nbsp;3 | url = http://fermatslasttheorem.blogspot.com/2005/05/fermats-last-theorem-proof-for-n3.html | access-date = 23 May 2009|date = 22 May 2005}}</ref> contained a major gap.<ref>Ribenboim, pp. 24–25</ref>{{sfn|Mordell|1921|pp=6–8|ps=}}{{sfn|Edwards|1996|pp=39–40|ps=}} However, since Euler himself had proved the lemma necessary to complete the proof in other work, he is generally credited with the first proof.{{sfn|Aczel|1996|p=44|ps=}}{{sfn|Edwards|1996|pp=40,52–54|ps=}}<ref>{{cite journal |author=J. J. Mačys |title=On Euler's hypothetical proof |journal=Mathematical Notes |year=2007 |volume=82 |issue=3–4 |pages=352–356 |doi=10.1134/S0001434607090088 |mr=2364600|s2cid=121798358 }}</ref> Independent proofs were published<ref>Ribenboim, pp. 33, 37–41</ref> by Kausler (1802),<ref name="Kausler_1802" >{{cite journal | author = Kausler CF | year = 1802 | title = Nova demonstratio theorematis nec summam, nec differentiam duorum cuborum cubum esse posse | journal =Novi Acta Academiae Scientiarum Imperialis Petropolitanae | volume = 13 | pages = 245–253}}</ref> Legendre (1823, 1830),<ref name="Legendre_1830" >{{cite book | author = Legendre AM | year = 1830 | title = Théorie des Nombres (Volume II) | edition = 3rd | publisher = Firmin Didot Frères | location = Paris| authorlink=Adrien-Marie Legendre }} Reprinted in 1955 by A. Blanchard (Paris).</ref><ref name="Legendre_1823" >{{cite journal | author = Legendre AM | year = 1823 | title = Recherches sur quelques objets d'analyse indéterminée, et particulièrement sur le théorème de Fermat | journal = Mémoires de l'Académie royale des sciences | volume = 6 | pages = 1–60| authorlink=Adrien-Marie Legendre }} Reprinted in 1825 as the "Second Supplément" for a printing of the 2nd edition of ''Essai sur la Théorie des Nombres'', Courcier (Paris). Also reprinted in 1909 in ''Sphinx-Oedipe'', '''4''', 97–128.</ref> Calzolari (1855),<ref>{{cite book | author = Calzolari L | year = 1855 | title = Tentativo per dimostrare il teorema di Fermat sull'equazione indeterminata x<sup>n</sup> + y<sup>n</sup> = z<sup>n</sup> | publisher = Ferrara}}</ref> [[Gabriel Lamé]] (1865),<ref>{{cite journal | author = Lamé G | year = 1865 | title = Étude des binômes cubiques ''x''<sup>3</sup> ± ''y''<sup>3</sup> | journal = [[Comptes rendus de l'Académie des sciences|Comptes rendus hebdomadaires des séances de l'Académie des Sciences]] | volume = 61 | pages = 921–924, 961–965| authorlink=Gabriel Lamé }}</ref> [[Peter Guthrie Tait]] (1872),<ref>{{cite journal | author = Tait PG | year = 1872 | title = Mathematical Notes | journal = Proceedings of the Royal Society of Edinburgh | volume = 7 | page = 144 | doi=10.1017/s0370164600041857| url = https://zenodo.org/record/1428704 | authorlink=Peter Guthrie Tait }}</ref> [[Siegmund Günther]] (1878),<ref>{{cite journal | last = Günther |first=S.|author-link=Siegmund Günther | year = 1878 | title = Ueber die unbestimmte Gleichung {{math|1=''x''<sup>''3''</sup> + ''y''<sup>''3''</sup> = ''a''<sup>''3''</sup>}} | journal = Sitzungsberichte der Königliche böhmische Gesellschaft der Wissenschaften in Prag |volume=jahrg. 1878-1880 | pages = 112–120|url=https://www.biodiversitylibrary.org/item/111872#page/150}}</ref> Gambioli (1901),<ref name="Gambioli_1901" >{{cite journal | author = Gambioli D | year = 1901 | title = Memoria bibliographica sull'ultimo teorema di Fermat | journal = Periodico di Matematiche | volume = 16 | pages = 145–192}}</ref> Krey (1909),<ref>{{cite journal|last=Krey|first=H.|year=1909|title=Neuer Beweis eines arithmetischen Satzes|journal=Mathematisch-Naturwissenschaftliche Blätter|volume=6|issue=12|pages=179–180|url=https://books.google.com/books?id=90Q6AQAAMAAJ&pg=RA2-PA179}}</ref> Rychlík (1910),<ref name="Rychlik_1910" >{{cite journal | author = Rychlik K | year = 1910 | title = On Fermat's last theorem for ''n''&nbsp;=&nbsp;4 and ''n''&nbsp;=&nbsp;3 (in Bohemian) | journal = Časopis Pro Pěstování Matematiky a Fysiky | volume = 39 | pages = 65–86| authorlink=Karel Rychlík }}</ref> Stockhaus (1910),<ref>{{cite book | author = Stockhaus H | year = 1910 | title = Beitrag zum Beweis des Fermatschen Satzes | publisher = Brandstetter | location = Leipzig}}</ref> Carmichael (1915),<ref>{{cite book | author = Carmichael RD | year = 1915 | title = Diophantine Analysis | publisher = Wiley | location = New York| authorlink=Robert Daniel Carmichael }}</ref> [[Johannes van der Corput]] (1915),<ref name="van_der_Corput_1915" >{{cite journal | author = van der Corput JG | year = 1915 | title = Quelques formes quadratiques et quelques équations indéterminées | journal = Nieuw Archief voor Wiskunde | volume = 11 | pages = 45–75| authorlink=Johannes van der Corput }}</ref> [[Axel Thue]] (1917),<ref>{{cite journal|last=Thue|first=Axel|author-link=Axel Thue|year=1917|title=Et bevis for at ligningen {{math|1=''A''<sup>3</sup> + ''B''<sup>3</sup> = ''C''<sup>3</sup>}} er unmulig i hele tal fra nul forskjellige tal ''A'', ''B'' og ''C''|journal=Archiv for Mathematik og Naturvidenskab|volume=34|issue=15|pages=3–7|url=https://books.google.com/books?id=xIFEAQAAMAAJ&pg=RA13-PA1}} Reprinted in ''Selected Mathematical Papers'' (1977), Oslo: Universitetsforlaget, pp. 555–559</ref> and Duarte (1944).<ref>{{cite journal | author = Duarte FJ | year = 1944 | title = Sobre la ecuación ''x''<sup>3</sup> + ''y''<sup>3</sup> + ''z''<sup>3</sup> = 0 | journal = Boletín de la Academia de Ciencias Físicas, Matemáticas y Naturales (Caracas) | volume = 8 | pages = 971–979}}</ref>

The case {{nowrap|1=''p'' = 5}} was proved<ref>{{cite web | author = Freeman L | title = Fermat's Last Theorem: Proof for ''n''&nbsp;=&nbsp;5 | url = http://fermatslasttheorem.blogspot.com/2005/10/fermats-last-theorem-proof-for-n5_28.html | access-date = 23 May 2009|date = 28 October 2005}}</ref> independently by Legendre and [[Peter Gustav Lejeune Dirichlet]] around 1825.<ref>Ribenboim, p. 49</ref>{{sfn|Mordell|1921|pp=8–9|ps=}}{{sfn|Aczel|1996|p=44|ps=}}<ref name="Singh, p. 106">Singh, p. 106</ref> Alternative proofs were developed<ref>Ribenboim, pp. 55–57</ref> by [[Carl Friedrich Gauss]] (1875, posthumous),<ref>{{cite book | author = Gauss CF | year = 1875 | chapter = Neue Theorie der Zerlegung der Cuben | title = Zur Theorie der complexen Zahlen, Werke, vol. II | edition = 2nd | publisher = Königl. Ges. Wiss. Göttingen | pages = 387–391| authorlink=Carl Friedrich Gauss }} (Published posthumously)</ref> Lebesgue (1843),<ref>{{cite journal | author = Lebesgue VA | year = 1843 | title = Théorèmes nouveaux sur l'équation indéterminée ''x''<sup>5</sup> + ''y''<sup>5</sup> = ''az''<sup>5</sup> | journal = [[Journal de Mathématiques Pures et Appliquées]] | volume = 8 | pages = 49–70| authorlink=Victor Lebesgue }}</ref> Lamé (1847),<ref>{{cite journal | author = Lamé G | year = 1847 | title = Mémoire sur la résolution en nombres complexes de l'équation ''A''<sup>5</sup> + ''B''<sup>5</sup> + ''C''<sup>5</sup> = 0 | journal = [[Journal de Mathématiques Pures et Appliquées]] | volume = 12 | pages = 137–171| authorlink=Gabriel Lamé }}</ref> Gambioli (1901),<ref name="Gambioli_1901"/><ref>{{cite journal | author = Gambioli D | date = 1903{{ndash}}1904 | title = Intorno all'ultimo teorema di Fermat | journal = Il Pitagora | volume = 10 | pages = 11–13, 41–42}}</ref> Werebrusow (1905),<ref>{{cite journal | author = Werebrusow AS | year = 1905 | title = On the equation ''x''<sup>5</sup> + ''y''<sup>5</sup> = ''Az''<sup>5</sup> ''(in Russian)'' | journal = Moskov. Math. Samml. | volume = 25 | pages = 466–473}}</ref>{{full citation needed|date=October 2017|reason=what is unabbreviated journal name?}} Rychlík (1910),<ref>{{cite journal | author = Rychlik K | year = 1910 | title = On Fermat's last theorem for ''n''&nbsp;=&nbsp;5 ''(in Bohemian)'' | journal = Časopis Pěst. Mat. | volume = 39 | pages = 185–195, 305–317| authorlink=Karel Rychlík }}</ref>{{dubious|date=October 2017|reason=it is unlikely that this article was published in the Bohemian language}}{{full citation needed|date=October 2017|reason=what is unabbreviated journal name?}} van der Corput (1915),<ref name="van_der_Corput_1915"/> and [[Guy Terjanian]] (1987).<ref>{{cite journal | author = Terjanian G | year = 1987 | title = Sur une question de V. A. Lebesgue | journal = Annales de l'Institut Fourier | volume = 37 | issue = 3 | pages = 19–37 | doi=10.5802/aif.1096| authorlink=Guy Terjanian | doi-access = free }}</ref>

The case {{nowrap|1=''p'' = 7}} was proved<ref>Ribenboim, pp. 57–63</ref>{{sfn|Mordell|1921|p=8|ps=}}{{sfn|Aczel|1996|p=44|ps=}}<ref name="Singh, p. 106"/> by Lamé in 1839.<ref name="Lame_1839" >{{cite journal | author = Lamé G | year = 1839 | title = Mémoire sur le dernier théorème de Fermat | journal = [[Comptes rendus de l'Académie des sciences|Comptes rendus hebdomadaires des séances de l'Académie des Sciences]] | volume = 9 | pages = 45–46| authorlink=Gabriel Lamé }}<br/>{{cite journal | author = Lamé G | year = 1840 | title = Mémoire d'analyse indéterminée démontrant que l'équation ''x''<sup>7</sup> + ''y''<sup>7</sup> = ''z''<sup>7</sup> est impossible en nombres entiers | journal = [[Journal de Mathématiques Pures et Appliquées]] | volume = 5 | pages = 195–211| authorlink=Gabriel Lamé }}</ref> His rather complicated proof was simplified in 1840 by Lebesgue,<ref>{{cite journal | author = Lebesgue VA | year = 1840 | title = Démonstration de l'impossibilité de résoudre l'équation ''x''<sup>7</sup> + ''y''<sup>7</sup> + ''z''<sup>7</sup> = 0 en nombres entiers | journal = [[Journal de Mathématiques Pures et Appliquées]] | volume = 5 | pages = 276–279, 348–349| authorlink=Victor Lebesgue }}</ref> and still simpler proofs<ref>{{cite web | author = Freeman L | title = Fermat's Last Theorem: Proof for ''n''&nbsp;=&nbsp;7 | url = http://fermatslasttheorem.blogspot.com/2006/01/fermats-last-theorem-proof-for-n7.html | access-date = 23 May 2009|date = 18 January 2006}}</ref> were published by [[Angelo Genocchi]] in 1864, 1874 and 1876.<ref>{{cite journal | author = Genocchi A | year = 1864 | title = Intorno all'equazioni ''x''<sup>7</sup> + ''y''<sup>7</sup> + ''z''<sup>7</sup> = 0 | journal = [[Annali di Matematica Pura ed Applicata]] | volume = 6 | pages = 287–288 | doi=10.1007/bf03198884| s2cid = 124916552 | authorlink=Angelo Genocchi | url = https://zenodo.org/record/1428472 }}<br/>{{cite journal | author = Genocchi A | year = 1874 | title = Sur l'impossibilité de quelques égalités doubles | journal = [[Comptes rendus de l'Académie des sciences|Comptes rendus hebdomadaires des séances de l'Académie des Sciences]] | volume = 78 | pages = 433–436| authorlink=Angelo Genocchi }}<br/>{{cite journal | author = Genocchi A | year = 1876 | title = Généralisation du théorème de Lamé sur l'impossibilité de l'équation ''x''<sup>7</sup> + ''y''<sup>7</sup> + ''z''<sup>7</sup> = 0 | journal = [[Comptes rendus de l'Académie des sciences|Comptes rendus hebdomadaires des séances de l'Académie des Sciences]] | volume = 82 | pages = 910–913| authorlink=Angelo Genocchi }}</ref> Alternative proofs were developed by Théophile Pépin (1876)<ref>{{cite journal | author = Pepin T | year = 1876 | title = Impossibilité de l'équation ''x''<sup>7</sup> + ''y''<sup>7</sup> + ''z''<sup>7</sup> = 0 | journal = [[Comptes rendus de l'Académie des sciences|Comptes rendus hebdomadaires des séances de l'Académie des Sciences]] | volume = 82 | pages = 676–679, 743–747| authorlink=Théophile Pépin }}</ref> and Edmond Maillet (1897).<ref>{{cite journal | author = Maillet E | year = 1897 | title = Sur l'équation indéterminée ''ax''<sup>''λ''<sup>''t''</sup></sup> + ''by''<sup>''λ''<sup>''t''</sup></sup> = ''cz''<sup>''λ''<sup>''t''</sup></sup> | journal = Association française pour l'avancement des sciences, St. Etienne, Compte Rendu de la 26me Session, deuxième partie | volume = 26 | pages = 156–168 |url=http://gallica.bnf.fr/ark:/12148/bpt6k201187c/f159.image| authorlink=Edmond Maillet }}</ref>

Fermat's Last Theorem was also proved for the exponents ''n''&nbsp;=&nbsp;6, 10, and 14. Proofs for {{nowrap|1=''n'' = 6}} were published by Kausler,<ref name="Kausler_1802"/> Thue,<ref>{{cite journal | author = Thue A | year = 1896 | title = Über die Auflösbarkeit einiger unbestimmter Gleichungen | journal = Det Kongelige Norske Videnskabers Selskabs Skrifter | volume = 7| authorlink=Axel Thue }} Reprinted in ''Selected Mathematical Papers'', pp. 19–30, Oslo: Universitetsforlaget (1977).</ref> Tafelmacher,<ref>{{cite journal | author = Tafelmacher WLA | year = 1897 | title = La ecuación ''x''<sup>3</sup> + ''y''<sup>3</sup> = ''z''<sup>2</sup>: Una demonstración nueva del teorema de fermat para el caso de las sestas potencias | journal = [[Anales de la Universidad de Chile]] | volume = 97 | pages = 63–80}}</ref> Lind,<ref>{{cite journal | author = Lind B | year = 1909 | title = Einige zahlentheoretische Sätze | journal = Archiv der Mathematik und Physik | volume = 15 | pages = 368–369}}</ref> Kapferer,<ref name="Kapferer_1913" >{{cite journal | author = Kapferer H | year = 1913 | title = Beweis des Fermatschen Satzes für die Exponenten 6 und 10 | journal = Archiv der Mathematik und Physik | volume = 21 | pages = 143–146}}</ref> Swift,<ref>{{cite journal | author = Swift E | year = 1914 | title = Solution to Problem 206 | journal = [[American Mathematical Monthly]] | volume = 21 | issue = 7 | pages = 238–239 | doi = 10.2307/2972379 | jstor = 2972379 }}</ref> and Breusch.<ref name="Breusch_1960" >{{cite journal | doi = 10.2307/3029800 | author = Breusch R | authorlink=Robert Breusch | year = 1960 | title = A simple proof of Fermat's last theorem for ''n''&nbsp;=&nbsp;6, ''n''&nbsp;=&nbsp;10 | journal = [[Mathematics Magazine]] | volume = 33 | issue = 5 | pages = 279–281 | jstor = 3029800}}</ref> Similarly, Dirichlet<ref>{{cite journal | author = Dirichlet PGL | year = 1832 | title = Démonstration du théorème de Fermat pour le cas des 14<sup>e</sup> puissances | journal = [[Crelle's Journal|Journal für die reine und angewandte Mathematik]] | volume = 9 | pages = 390–393| authorlink=Peter Gustav Lejeune Dirichlet }} Reprinted in ''Werke'', vol. I, pp. 189–194, Berlin: G. Reimer (1889); reprinted New York:Chelsea (1969).</ref> and Terjanian<ref>{{cite journal | author = Terjanian G | year = 1974 | title= L'équation ''x''<sup>14</sup> + ''y''<sup>14</sup> = ''z''<sup>14</sup> en nombres entiers | journal = Bulletin des Sciences Mathématiques |series=Série 2 | volume = 98 | pages = 91–95| authorlink=Guy Terjanian }}</ref> each proved the case ''n''&nbsp;=&nbsp;14, while Kapferer<ref name="Kapferer_1913"/> and Breusch<ref name="Breusch_1960"/> each proved the case ''n''&nbsp;=&nbsp;10. Strictly speaking, these proofs are unnecessary, since these cases follow from the proofs for ''n''&nbsp;=&nbsp;3, 5, and 7, respectively. Nevertheless, the reasoning of these even-exponent proofs differs from their odd-exponent counterparts. Dirichlet's proof for ''n''&nbsp;=&nbsp;14 was published in 1832, before Lamé's 1839 proof for {{nowrap|1=''n'' = 7}}.{{sfn|Edwards|1996|pp=73–74|ps=}}

All proofs for specific exponents used Fermat's technique of [[infinite descent]],{{citation needed|reason=this assertion requires a thorough review of all the existing literature, or a reliable source making the assertion|date=January 2015}} either in its original form, or in the form of descent on elliptic curves or abelian varieties. The details and auxiliary arguments, however, were often ''ad hoc'' and tied to the individual exponent under consideration.{{sfn|Edwards|1996|p=74|ps=}} Since they became ever more complicated as ''p'' increased, it seemed unlikely that the general case of Fermat's Last Theorem could be proved by building upon the proofs for individual exponents.{{sfn|Edwards|1996|p=74|ps=}} Although some general results on Fermat's Last Theorem were published in the early 19th century by [[Niels Henrik Abel]] and [[Peter Barlow (mathematician)|Peter Barlow]],{{sfn|Dickson|1919|p=733|ps=}}<ref>{{cite book | author = Ribenboim P | year = 1979 | title = 13 Lectures on Fermat's Last Theorem | publisher = Springer Verlag | location = New York | isbn = 978-0-387-90432-0 | pages = 51–54| authorlink=Paulo Ribenboim }}</ref> the first significant work on the general theorem was done by [[Sophie Germain]].<ref>Singh, pp. 97–109</ref>

=== Early modern breakthroughs ===
==== Sophie Germain ====
In the early 19th century, [[Sophie Germain]] developed several novel approaches to prove Fermat's Last Theorem for all exponents.<ref name="Laubenbacher_2007" >{{cite web | vauthors = Laubenbacher R, Pengelley D | year = 2007 | title = Voici ce que j'ai trouvé: Sophie Germain's grand plan to prove Fermat's Last Theorem | url = http://www.math.nmsu.edu/%7Edavidp/germain.pdf | access-date = 19 May 2009 | archive-url = https://web.archive.org/web/20130405163013/http://www.math.nmsu.edu/%7Edavidp/germain.pdf | archive-date = 5 April 2013 | url-status = dead }}</ref> First, she defined a set of auxiliary primes ''θ'' constructed from the prime exponent ''p'' by the equation {{nowrap|1=''θ'' = 2''hp'' + 1}}, where ''h'' is any integer not divisible by three. She showed that, if no integers raised to the ''p''th power were adjacent modulo ''θ'' (the ''non-consecutivity condition''), then ''θ'' must divide the product ''xyz''. Her goal was to use [[mathematical induction]] to prove that, for any given ''p'', infinitely many auxiliary primes ''θ'' satisfied the non-consecutivity condition and thus divided ''xyz''; since the product ''xyz'' can have at most a finite number of prime factors, such a proof would have established Fermat's Last Theorem. Although she developed many techniques for establishing the non-consecutivity condition, she did not succeed in her strategic goal. She also worked to set lower limits on the size of solutions to Fermat's equation for a given exponent ''p'', a modified version of which was published by [[Adrien-Marie Legendre]]. As a byproduct of this latter work, she proved [[Sophie Germain's theorem]], which verified the first case of Fermat's Last Theorem (namely, the case in which ''p'' does not divide ''xyz'') for every odd prime exponent less than 270,<ref name="Laubenbacher_2007"/>{{sfn|Aczel|1996|p=57|ps=}} and for all primes ''p'' such that at least one of {{nowrap|2''p'' + 1}}, {{nowrap|4''p'' + 1}}, {{nowrap|8''p'' + 1}}, {{nowrap|10''p'' + 1}}, {{nowrap|14''p'' + 1}} and {{nowrap|16''p'' + 1}} is prime (specially, the primes ''p'' such that {{nowrap|2''p'' + 1}} is prime are called [[Sophie Germain prime]]s). Germain tried unsuccessfully to prove the first case of Fermat's Last Theorem for all even exponents, specifically for {{nowrap|1=''n'' = 2''p''}}, which was proved by [[Guy Terjanian]] in 1977.<ref>{{cite journal|last=Terjanian|first=G.|year=1977|title=Sur l'équation ''x''<sup>2''p''</sup> + ''y''<sup>2''p''</sup> = ''z''<sup>2''p''</sup> |journal=Comptes Rendus de l'Académie des Sciences, Série A-B|volume=285|pages=973–975}}</ref> In 1985, [[Leonard Adleman]], [[Roger Heath-Brown]] and [[Étienne Fouvry]] proved that the first case of Fermat's Last Theorem holds for infinitely many odd primes ''p''.<ref>{{cite journal |vauthors=Adleman LM, Heath-Brown DR |date=June 1985 |title = The first case of Fermat's last theorem | journal = Inventiones Mathematicae | volume = 79 | issue = 2 |pages = 409–416 | publisher = Springer | location = Berlin | doi=10.1007/BF01388981 | bibcode=1985InMat..79..409A|s2cid=122537472 }}</ref>

==== Ernst Kummer and the theory of ideals ====
In 1847, [[Gabriel Lamé]] outlined a proof of Fermat's Last Theorem based on factoring the equation {{math|1={{itco|''x''}}{{sup|''p''}} + {{itco|''y''}}{{sup|''p''}} = {{itco|''z''}}{{sup|''p''}}}} in [[complex number]]s, specifically the [[cyclotomic field]] based on the [[root of unity|roots of the number 1]]. His proof failed, however, because it assumed incorrectly that such complex numbers can be [[unique factorization|factored uniquely]] into primes, similar to integers. This gap was pointed out immediately by [[Joseph Liouville]], who later read a paper that demonstrated this failure of unique factorisation, written by [[Ernst Kummer]].

Kummer set himself the task of determining whether the [[cyclotomic field]] could be generalized to include new prime numbers such that unique factorisation was restored. He succeeded in that task by developing the [[ideal number]]s.

(It is often stated that Kummer was led to his "ideal complex numbers" by his interest in Fermat's Last Theorem; there is even a story often told that Kummer, like [[Gabriel Lamé|Lamé]], believed he had proven Fermat's Last Theorem until [[Peter Gustav Lejeune Dirichlet|Lejeune Dirichlet]] told him his argument relied on unique factorization; but the story was first told by [[Kurt Hensel]] in 1910 and the evidence indicates it likely derives from a confusion by one of Hensel's sources. [[Harold Edwards (mathematician)|Harold Edwards]] said the belief that Kummer was mainly interested in Fermat's Last Theorem "is surely mistaken".{{sfn|Edwards|1996|p=79|ps=}} See [[Ideal number#History|the history of ideal numbers]].)

Using the general approach outlined by Lamé, Kummer proved both cases of Fermat's Last Theorem for all [[regular prime|regular prime numbers]]. However, he could not prove the theorem for the exceptional primes (irregular primes) that [[Regular prime#Siegel's conjecture|conjecturally occur approximately 39% of the time]]; the only irregular primes below 270 are 37, 59, 67, 101, 103, 131, 149, 157, 233, 257 and 263.

==== Mordell conjecture ====
In the 1920s, [[Louis Mordell]] posed a conjecture that implied that Fermat's equation has at most a finite number of nontrivial primitive integer solutions, if the exponent ''n'' is greater than two.{{sfn|Aczel|1996|pp=84–88|ps=}}<ref>Singh, pp. 232–234</ref> This conjecture was proved in 1983 by [[Gerd Faltings]],<ref>{{cite journal | author = Faltings G | year = 1983 | title = Endlichkeitssätze für abelsche Varietäten über Zahlkörpern |journal = Inventiones Mathematicae | volume = 73 | issue = 3 |pages = 349–366 |doi=10.1007/BF01388432|bibcode = 1983InMat..73..349F | s2cid = 121049418 | authorlink=Gerd Faltings }}</ref> and is now known as [[Faltings's theorem]].

==== Computational studies ====
In the latter half of the 20th century, computational methods were used to extend Kummer's approach to the irregular primes. In 1954, [[Harry Vandiver]] used a [[SWAC (computer)|SWAC computer]] to prove Fermat's Last Theorem for all primes up to 2521.<ref>{{cite book | author = Ribenboim P | year = 1979 | title = 13 Lectures on Fermat's Last Theorem | publisher = Springer Verlag | location = New York | isbn = 978-0-387-90432-0 | page = 202| authorlink=Paulo Ribenboim }}</ref> By 1978, [[Samuel S. Wagstaff, Jr.|Samuel Wagstaff]] had extended this to all primes less than 125,000.<ref>{{cite journal | author = Wagstaff SS Jr. | year = 1978 | title = The irregular primes to 125000 | journal = Mathematics of Computation | volume = 32 | pages = 583–591 | doi = 10.2307/2006167 | issue = 142 | publisher = American Mathematical Society | jstor = 2006167| authorlink=Samuel S. Wagstaff, Jr }} [https://www.ams.org/journals/mcom/1978-32-142/S0025-5718-1978-0491465-4/S0025-5718-1978-0491465-4.pdf (PDF)] {{webarchive |url=https://web.archive.org/web/20121024112422/http://www.ams.org/journals/mcom/1978-32-142/S0025-5718-1978-0491465-4/S0025-5718-1978-0491465-4.pdf |date=24 October 2012 }}</ref> By 1993, Fermat's Last Theorem had been proved for all primes less than four million.<ref name=":0">{{cite journal|vauthors=Buhler J, Crandell R, Ernvall R, Metsänkylä T|year=1993|title=Irregular primes and cyclotomic invariants to four million|url=https://www.ams.org/journals/mcom/1993-61-203/S0025-5718-1993-1197511-5/home.html|journal=Mathematics of Computation|publisher=American Mathematical Society|volume=61|issue=203|pages=151–153|bibcode=1993MaCom..61..151B|doi=10.2307/2152942|jstor=2152942|doi-access=free}}</ref>

However, despite these efforts and their results, no proof existed of Fermat's Last Theorem. Proofs of individual exponents by their nature could never prove the ''general'' case: even if all exponents were verified up to an extremely large number X, a higher exponent beyond X might still exist for which the claim was not true. (This had been the case with some other past conjectures, such as with [[Skewes' number]], and it could not be ruled out in this conjecture.)

=== Connection with elliptic curves ===
The strategy that ultimately led to a successful proof of Fermat's Last Theorem arose from the "astounding"<ref name="Singh">Fermat's Last Theorem, Simon Singh, 1997, {{isbn|1-85702-521-0}}</ref>{{rp|211}} [[modularity theorem|Taniyama–Shimura–Weil conjecture]], proposed around 1955—which many mathematicians believed would be near to impossible to prove,<ref name="Singh"/>{{rp|223}} and was linked in the 1980s by [[Gerhard Frey]], [[Jean-Pierre Serre]] and [[Ken Ribet]] to Fermat's equation. By accomplishing a partial proof of this conjecture in 1994, [[Andrew Wiles]] ultimately succeeded in proving Fermat's Last Theorem, as well as leading the way to a full proof by others of what is now known as the [[modularity theorem]].

==== Taniyama–Shimura–Weil conjecture ====
{{main|Modularity theorem}}

Around 1955, Japanese mathematicians [[Goro Shimura]] and [[Yutaka Taniyama]] observed a possible link between two apparently completely distinct branches of mathematics, [[elliptic curve]]s and [[modular form]]s. The resulting [[modularity theorem]] (at the time known as the Taniyama–Shimura conjecture) states that every elliptic curve is [[modular elliptic curve|modular]], meaning that it can be associated with a unique [[modular form]].

The link was initially dismissed as unlikely or highly speculative, but was taken more seriously when number theorist [[André Weil]] found evidence supporting it, though not proving it; as a result the conjecture was often known as the Taniyama–Shimura–Weil conjecture.<ref name="Singh"/>{{rp|211–215}}

Even after gaining serious attention, the conjecture was seen by contemporary mathematicians as extraordinarily difficult or perhaps inaccessible to proof.<ref name="Singh"/>{{rp|203–205, 223, 226}} For example, Wiles's doctoral supervisor [[John H. Coates|John Coates]] states that it seemed "impossible to actually prove",<ref name="Singh"/>{{rp|226}} and Ken Ribet considered himself "one of the vast majority of people who believed [it] was completely inaccessible", adding that "Andrew Wiles was probably one of the few people on earth who had the audacity to dream that you can actually go and prove [it]."<ref name="Singh"/>{{rp|223}}

==== Ribet's theorem for Frey curves ====
{{main|Frey curve|Ribet's theorem}}

In 1984, [[Gerhard Frey]] noted a link between Fermat's equation and the modularity theorem, then still a conjecture. If Fermat's equation had any solution {{nowrap|(''a'', ''b'', ''c'')}} for exponent {{nowrap|''p'' > 2}}, then it could be shown that the semi-stable [[elliptic curve]] (now known as a [[Frey curve|Frey-Hellegouarch]]<ref group=note>This elliptic curve was first suggested in the 1960s by {{Interlanguage link|Yves Hellegouarch|de}}, but he did not call attention to its non-modularity. For more details, see {{Cite book | last=Hellegouarch | first=Yves | title=Invitation to the Mathematics of Fermat-Wiles | publisher=Academic Press | year=2001 | isbn=978-0-12-339251-0 }}</ref>)
: ''y''<sup>2</sup> = {{nowrap|''x''(''x'' − ''a''<sup>''p''</sup>)(''x'' + ''b''<sup>''p''</sup>)}}
would have such unusual properties that it was unlikely to be modular.<ref>{{cite journal | author = Frey G | year = 1986 |title = Links between stable elliptic curves and certain diophantine equations | journal = Annales Universitatis Saraviensis. Series Mathematicae. | volume = 1 | pages = 1–40| authorlink=Gerhard Frey }}</ref> This would conflict with the modularity theorem, which asserted that all elliptic curves are modular. As such, Frey observed that a proof of the Taniyama–Shimura–Weil conjecture might also simultaneously prove Fermat's Last Theorem.<ref>Singh, pp. 194–198</ref>{{sfn|Aczel|1996||pp=109–114|ps=}} By [[contraposition]], a ''disproof'' or refutation of Fermat's Last Theorem would disprove the Taniyama–Shimura–Weil conjecture.

In plain English, Frey had shown that, if this intuition about his equation was correct, then any set of four numbers (''a'', ''b'', ''c'', ''n'') capable of disproving Fermat's Last Theorem, could also be used to disprove the Taniyama–Shimura–Weil conjecture. Therefore, if the latter were true, the former could not be disproven, and would also have to be true.

Following this strategy, a proof of Fermat's Last Theorem required two steps. First, it was necessary to prove the modularity theorem, or at least to prove it for the types of elliptical curves that included Frey's equation (known as [[semistable elliptic curve]]s). This was widely believed inaccessible to proof by contemporary mathematicians.<ref name="Singh"/>{{rp|203–205, 223, 226}} Second, it was necessary to show that Frey's intuition was correct: that if an elliptic curve were constructed in this way, using a set of numbers that were a solution of Fermat's equation, the resulting elliptic curve could not be modular. Frey showed that this was ''plausible'' but did not go as far as giving a full proof. The missing piece (the so-called "[[epsilon conjecture]]", now known as [[Ribet's theorem]]) was identified by [[Jean-Pierre Serre]] who also gave an almost-complete proof<!-- see introduction of Ribet's paper --> and the link suggested by Frey was finally proved in 1986 by [[Ken Ribet]].<ref name="ribet90">{{cite journal|last=Ribet|first=Ken|authorlink=Ken Ribet|title=On modular representations of Gal({{overline|'''Q'''}}/'''Q''') arising from modular forms|journal=Inventiones Mathematicae|volume=100|year=1990|issue=2|pages=431–476|doi=10.1007/BF01231195|mr=1047143|url=http://math.berkeley.edu/~ribet/Articles/invent_100.pdf|bibcode=1990InMat.100..431R|hdl=10338.dmlcz/147454|s2cid=120614740}}</ref>

Following Frey, Serre and Ribet's work, this was where matters stood:

* Fermat's Last Theorem needed to be proven for all exponents ''n'' that were prime numbers.
* The modularity theorem—if proved for semi-stable elliptic curves—would mean that all semistable elliptic curves ''must'' be modular.
* Ribet's theorem showed that any solution to Fermat's equation for a prime number could be used to create a semistable elliptic curve that ''could not'' be modular;
* The only way that both of these statements could be true, was if ''no'' solutions existed to Fermat's equation (because then no such curve could be created), which was what Fermat's Last Theorem said. As Ribet's Theorem was already proved, this meant that a proof of the modularity theorem would automatically prove Fermat's Last theorem was true as well.

=== Wiles's general proof ===
{{Main|Andrew Wiles|Wiles's proof of Fermat's Last Theorem}}
[[File:Andrew wiles1-3.jpg|thumb|upright=0.65|British mathematician [[Andrew Wiles]]]]

Ribet's proof of the [[epsilon conjecture]] in 1986 accomplished the first of the two goals proposed by Frey. Upon hearing of Ribet's success, [[Andrew Wiles]], an English mathematician with a childhood fascination with Fermat's Last Theorem, and who had worked on elliptic curves, decided to commit himself to accomplishing the second half: proving a special case of the [[modularity theorem]] (then known as the Taniyama–Shimura conjecture) for semistable elliptic curves.<ref>Singh, p. 205</ref>{{sfn|Aczel|1996|pp=117–118|ps=}}

Wiles worked on that task for six years in near-total secrecy, covering up his efforts by releasing prior work in small segments as separate papers and confiding only in his wife.<ref name="Singh"/>{{rp|229–230}} His initial study suggested [[Mathematical proof|proof]] by [[mathematical induction|induction]],<ref name="Singh"/>{{rp|230–232, 249–252}} and he based his initial work and first significant breakthrough on [[Galois theory]]<ref name="Singh"/>{{rp|251–253, 259}} before switching to an attempt to extend [[Iwasawa theory|horizontal Iwasawa theory]] for the inductive argument around 1990–91 when it seemed that there was no existing approach adequate to the problem.<ref name="Singh"/>{{rp|258–259}} However, by mid-1991, Iwasawa theory also seemed to not be reaching the central issues in the problem.<ref name="Singh"/>{{rp|259–260}}<ref>Singh, pp. 237–238</ref>{{sfn|Aczel|1996|pp=121–122|ps=}} In response, he approached colleagues to seek out any hints of cutting-edge research and new techniques, and discovered an [[Euler system]] recently developed by [[Victor Kolyvagin]] and [[Matthias Flach (mathematician)|Matthias Flach]] that seemed "tailor made" for the inductive part of his proof.<ref name="Singh"/>{{rp|260–261}} Wiles studied and extended this approach, which worked. Since his work relied extensively on this approach, which was new to mathematics and to Wiles, in January 1993 he asked his Princeton colleague, [[Nick Katz]], to help him check his reasoning for subtle errors. Their conclusion at the time was that the techniques Wiles used seemed to work correctly.<ref name="Singh"/>{{rp|261–265}}<ref>Singh, pp. 239–243</ref>{{sfn|Aczel|1996|pp=122–125|ps=}}

By mid-May 1993, Wiles was ready to tell his wife he thought he had solved the proof of Fermat's Last Theorem,<ref name="Singh"/>{{rp|265}} and by June he felt sufficiently confident to present his results in three lectures delivered on 21–23 June 1993 at the [[Isaac Newton Institute for Mathematical Sciences]].<ref>Singh, pp. 244–253</ref>{{sfn|Aczel|1996|pp=1–4,126–128|ps=}} Specifically, Wiles presented his proof of the Taniyama–Shimura conjecture for semistable elliptic curves; together with Ribet's proof of the epsilon conjecture, this implied Fermat's Last Theorem. However, it became apparent during [[peer review]] that a critical point in the proof was incorrect. It contained an error in a bound on the order of a particular [[group (mathematics)|group]]. The error was caught by several mathematicians refereeing Wiles's manuscript including Katz (in his role as reviewer),{{sfn|Aczel|1996|pp=128–130|ps=}} who alerted Wiles on 23 August 1993.<ref>Singh, p. 257</ref>

The error would not have rendered his work worthless: each part of Wiles's work was highly significant and innovative by itself, as were the many developments and techniques he had created in the course of his work, and only one part was affected.<ref name="Singh"/>{{rp|289, 296–297}} However, without this part proved, there was no actual proof of Fermat's Last Theorem. Wiles spent almost a year trying to repair his proof, initially by himself and then in collaboration with his former student [[Richard Taylor (mathematician)|Richard Taylor]], without success.<ref name="sept1994">Singh, pp. 269–277</ref><ref>[https://www.nytimes.com/1994/06/28/science/a-year-later-snag-persists-in-math-proof.html A Year Later, Snag Persists In Math Proof] 28 June 1994</ref><ref>[https://www.nytimes.com/1994/07/03/weekinreview/june-26-july-2-a-year-later-fermat-s-puzzle-is-still-not-quite-qed.html 26 June – 2 July; A Year Later Fermat's Puzzle Is Still Not Quite Q.E.D.] 3 July 1994</ref> By the end of 1993, rumours had spread that under scrutiny, Wiles's proof had failed, but how seriously was not known. Mathematicians were beginning to pressure Wiles to disclose his work whether it was complete or not, so that the wider community could explore and use whatever he had managed to accomplish. But instead of being fixed, the problem, which had originally seemed minor, now seemed very significant, far more serious, and less easy to resolve.<ref>Singh, pp. 175–185</ref>

Wiles states that on the morning of 19 September 1994, he was on the verge of giving up and was almost resigned to accepting that he had failed, and to publishing his work so that others could build on it and fix the error. He adds that he was having a final look to try and understand the fundamental reasons why his approach could not be made to work, when he had a sudden insight: that the specific reason why the Kolyvagin–Flach approach would not work directly {{em|also}} meant that his original attempts using [[Iwasawa theory]] could be made to work, if he strengthened it using his experience gained from the Kolyvagin–Flach approach. Fixing one approach with tools from the other approach would resolve the issue for all the cases that were not already proven by his refereed paper.<ref name="sept1994"/>{{sfn|Aczel|1996|pp=132–134|ps=}} He described later that Iwasawa theory and the Kolyvagin–Flach approach were each inadequate on their own, but together they could be made powerful enough to overcome this final hurdle.<ref name="sept1994"/>

{{blockquote|I was sitting at my desk examining the Kolyvagin–Flach method. It wasn't that I believed I could make it work, but I thought that at least I could explain why it didn't work. Suddenly I had this incredible revelation. I realised that, the Kolyvagin–Flach method wasn't working, but it was all I needed to make my original Iwasawa theory work from three years earlier. So out of the ashes of Kolyvagin–Flach seemed to rise the true answer to the problem. It was so indescribably beautiful; it was so simple and so elegant. I couldn't understand how I'd missed it and I just stared at it in disbelief for twenty minutes. Then during the day I walked around the department, and I'd keep coming back to my desk looking to see if it was still there. It was still there. I couldn't contain myself, I was so excited. It was the most important moment of my working life. Nothing I ever do again will mean as much.|Andrew Wiles, as quoted by Simon Singh<ref>Singh p. 186–187 (text condensed)</ref>}}

On 24 October 1994, Wiles submitted two manuscripts, "Modular elliptic curves and Fermat's Last Theorem"<ref>{{cite journal|last=Wiles|first=Andrew|authorlink=Andrew Wiles|year=1995|title=Modular elliptic curves and Fermat's Last Theorem|url=http://math.stanford.edu/~lekheng/flt/wiles.pdf|url-status=dead|journal=[[Annals of Mathematics]]|volume=141|issue=3|pages=443–551|doi=10.2307/2118559|jstor=2118559|oclc=37032255|archive-url=https://web.archive.org/web/20030628220453/http://math.stanford.edu/~lekheng/flt/wiles.pdf|archive-date=28 June 2003}}</ref><ref>{{Cite web|url=http://scienzamedia.uniroma2.it/~eal/Wiles-Fermat.pdf|title=Modular elliptic curves and Fermat's Last Theorem}}</ref> and "Ring theoretic properties of certain Hecke algebras",<ref>{{cite journal|author=[[Richard Taylor (mathematician)|Taylor R]], [[Andrew Wiles|Wiles A]] |year=1995 |journal=[[Annals of Mathematics]] |title=Ring theoretic properties of certain Hecke algebras |volume=141 |issue=3 |pages=553–572 |oclc=37032255 |url=http://www.math.harvard.edu/~rtaylor/hecke.ps |doi=10.2307/2118560 |jstor=2118560 |url-status=dead |archive-url=https://web.archive.org/web/20011127181043/http://www.math.harvard.edu/~rtaylor/hecke.ps |archive-date=27 November 2001 }}</ref> the second of which was co-authored with Taylor and proved that certain conditions were met that were needed to justify the corrected step in the main paper. The two papers were vetted and published as the entirety of the May 1995 issue of the ''[[Annals of Mathematics]]''. The proof's method of identification of a deformation ring with a Hecke algebra (now referred to as an ''R=T theorem'') to prove modularity lifting theorems has been an influential development in [[algebraic number theory]].

These papers established the modularity theorem for semistable elliptic curves, the last step in proving Fermat's Last Theorem, 358 years after it was conjectured.

=== Subsequent developments ===
The full Taniyama–Shimura–Weil conjecture was finally proved by Diamond (1996),<ref name="Diamond-1996"/> Conrad et al. (1999),<ref name="Conrad-etal-1999"/> and Breuil et al. (2001)<ref name="Breuil-etal-2001"/> who, building on Wiles's work, incrementally chipped away at the remaining cases until the full result was proved. The now fully proved conjecture became known as the [[modularity theorem]].

Several other theorems in number theory similar to Fermat's Last Theorem also follow from the same reasoning, using the modularity theorem. For example: no cube can be written as a sum of two coprime ''n''th powers, {{nowrap|''n'' ≥ 3}}. (The case {{nowrap|1=''n'' = 3}} was already known by [[Euler]].)

== Relationship to other problems and generalizations ==
Fermat's Last Theorem considers solutions to the Fermat equation: {{math|1=''a''<sup>''n''</sup> + ''b''<sup>''n''</sup> = ''c''<sup>''n''</sup>}} with positive integers {{math|''a''}}, {{math|''b''}}, and {{math|''c''}} and an integer {{math|''n''}} greater than 2. There are several generalizations of the Fermat equation to more general equations that allow the exponent {{math|''n''}} to be a negative integer or rational, or to consider three different exponents.

=== Generalized Fermat equation ===
The generalized Fermat equation generalizes the statement of Fermat's last theorem by considering positive integer solutions ''a'', ''b'', ''c'', ''m'', ''n'', ''k'' satisfying<ref name="princeton-companion">{{cite book |title=The Princeton Companion to Mathematics |url=https://archive.org/details/princetoncompanio00gowe |last1=Barrow-Green |first1=June |last2=Leader |first2=Imre |last3=Gowers |first3=Timothy |pages=361–362 |year=2008 |publisher=Princeton University Press|isbn=9781400830398 }}</ref>
{{NumBlk|:|<math>a^m + b^n = c^k.</math>|{{EquationRef|1}}}}
In particular, the exponents ''m'', ''n'', ''k'' need not be equal, whereas Fermat's last theorem considers the case {{math|1=''m'' = ''n'' = ''k''.}}

The [[Beal conjecture]], also known as the Mauldin conjecture<ref>{{cite web |url = http://www.primepuzzles.net/puzzles/puzz_559.htm | title = Mauldin / Tijdeman-Zagier Conjecture | publisher = Prime Puzzles | access-date = 1 October 2016}}</ref> and the Tijdeman-Zagier conjecture,<ref name=Elkies>{{cite journal |last=Elkies| first = Noam D. | title=The ABC's of Number Theory | journal = The Harvard College Mathematics Review | year=2007 | volume=1 | issue = 1 | url=http://dash.harvard.edu/bitstream/handle/1/2793857/Elkies%20-%20ABCs%20of%20Number%20Theory.pdf?sequence=2}}</ref><ref>{{cite journal |journal=  Moscow Mathematical Journal|volume=4 |year=2004 |title=Open Diophantine Problems |pages=245–305 |author=Michel Waldschmidt |doi=10.17323/1609-4514-2004-4-1-245-305 |arxiv=math/0312440|s2cid=11845578 }}</ref><ref name="PrimeNumbers">{{cite book |title=Prime Numbers: A Computational Perspective |last1=Crandall |first1=Richard |last2=Pomerance |first2=Carl |year=2000 |isbn=978-0387-25282-7 |publisher=Springer |page=417}}</ref> states that there are no solutions to the generalized Fermat equation in positive integers ''a'', ''b'', ''c'', ''m'', ''n'', ''k'' with ''a'', ''b'', and ''c'' being pairwise coprime and all of ''m'', ''n'', ''k'' being greater than 2.<ref>{{cite web| url=https://www.ams.org/profession/prizes-awards/ams-supported/beal-conjecture | title=Beal Conjecture | publisher=American Mathematical Society | access-date=21 August 2016}}</ref>

The [[Fermat–Catalan conjecture]] generalizes Fermat's last theorem with the ideas of the [[Catalan conjecture]].<ref>{{cite journal |title=A new generalization of Fermat's Last Theorem |last1=Cai |first1=Tianxin |last2=Chen |first2=Deyi |last3=Zhang |first3=Yong |journal=Journal of Number Theory |volume=149 |year=2015 |pages=33–45|doi=10.1016/j.jnt.2014.09.014 |arxiv=1310.0897 |s2cid=119732583 }}</ref><ref>{{cite journal |title=A Cyclotomic Investigation of the Catalan–Fermat Conjecture |last=Mihailescu |first=Preda |journal=Mathematica Gottingensis |year=2007}}</ref> The conjecture states that the generalized Fermat equation has only ''finitely many'' solutions (''a'', ''b'', ''c'', ''m'', ''n'', ''k'') with distinct triplets of values (''a''<sup>''m''</sup>, ''b''<sup>''n''</sup>, ''c''<sup>''k''</sup>), where ''a'', ''b'', ''c'' are positive coprime integers and ''m'', ''n'', ''k'' are positive integers satisfying

{{NumBlk|:|<math>\frac{1}{m} + \frac{1}{n} + \frac{1}{k} < 1.</math>|{{EquationRef|2}}}}

The statement is about the finiteness of the set of solutions because there are 10 [[Fermat–Catalan conjecture#Known solutions|known solutions]].<ref name="princeton-companion"/>

=== Inverse Fermat equation ===
When we allow the exponent {{math|''n''}} to be the reciprocal of an integer, i.e. {{math|1=''n'' = 1/''m''}} for some integer {{math|''m''}}, we have the inverse Fermat equation {{nowrap|1=''a''<sup>1/''m''</sup> + ''b''<sup>1/''m''</sup> = ''c''<sup>1/''m''</sup>}}.
All solutions of this equation were computed by [[Hendrik Lenstra]] in 1992.<ref>{{cite journal | author = Lenstra Jr. H.W. | authorlink=Hendrik Lenstra | year = 1992 | title = On the inverse Fermat equation | journal = Discrete Mathematics | volume = 106–107 | pages = 329–331 | doi = 10.1016/0012-365x(92)90561-s | doi-access =  }}</ref> In the case in which the ''m''th roots are required to be real and positive, all solutions are given by<ref>{{cite journal | author = Newman M | year = 1981 | title = A radical diophantine equation | doi = 10.1016/0022-314x(81)90040-8 | journal = [[Journal of Number Theory]] | volume = 13 | issue = 4| pages = 495–498 | doi-access = free }}</ref>
: <math>a=rs^m</math>
: <math>b=rt^m</math>
: <math>c=r(s+t)^m</math>
for positive integers ''r'', ''s'', ''t'' with ''s'' and ''t'' coprime.

=== Rational exponents ===
For the Diophantine equation {{nowrap|1=''a''<sup>''n''/''m''</sup> + ''b''<sup>''n''/''m''</sup> = ''c''<sup>''n''/''m''</sup>}} with ''n'' not equal to 1, Bennett, Glass, and Székely proved in 2004 for {{nowrap|''n'' > 2}}, that if ''n'' and ''m'' are coprime, then there are integer solutions if and only if 6 divides ''m'', and {{nowrap|''a''<sup>1/''m''</sup>}}, {{nowrap|''b''<sup>1/''m''</sup>}}, and {{nowrap|''c''<sup>1/''m''</sup>}} are different complex 6th roots of the same real number.<ref>
{{cite journal
 | last1 = Bennett | first1 = Curtis D.
 | last2 = Glass | first2 = A. M. W.
 | last3 = Székely | first3 = Gábor J.
 | doi = 10.2307/4145241
 | issue = 4
 | journal = American Mathematical Monthly
 | mr = 2057186
 | pages = 322–329
 | title = Fermat's last theorem for rational exponents
 | volume = 111
 | year = 2004| jstor = 4145241
}}</ref>

=== Negative integer exponents ===
==== ''n'' = −1 ====
All primitive integer solutions (i.e., those with no prime factor common to all of ''a'', ''b'', and ''c'') to the [[optic equation]] {{nowrap|1=''a''<sup>−1</sup> + ''b''<sup>−1</sup> = ''c''<sup>−1</sup>}} can be written as{{sfn|Dickson|1919|pp=688–691|ps=}}
: <math>a = mk + m^2,</math>
: <math>b = mk + k^2,</math>
: <math>c = mk</math>
for positive, coprime integers ''m'', ''k''.

==== ''n'' = −2 ====
The case {{nowrap|1=''n'' = −2}} also has an infinitude of solutions, and these have a geometric interpretation in terms of [[Integer triangle#Pythagorean triangles with integer altitude from the hypotenuse|right triangles with integer sides and an integer altitude to the hypotenuse]].<ref>{{cite journal |last=Voles |first=Roger |title=Integer solutions of ''a''<sup>−2</sup>&nbsp;+&nbsp;''b''<sup>−2</sup>&nbsp;=&nbsp;''d''<sup>−2</sup> |journal=[[Mathematical Gazette]] |volume=83 |issue=497 |date=July 1999 |pages=269–271|doi=10.2307/3619056 |jstor=3619056 |s2cid=123267065 }}</ref><ref>{{cite journal |last=Richinick |first=Jennifer |title=The upside-down Pythagorean Theorem |journal=[[Mathematical Gazette]] |volume=92 |date=July 2008 |pages=313–317|doi=10.1017/S0025557200183275 |s2cid=125989951 }}</ref> All primitive solutions to {{nowrap|1=''a''<sup>−2</sup> + ''b''<sup>−2</sup> = ''d''<sup>−2</sup>}} are given by
: <math>a = (v^2 - u^2)(v^2 + u^2),</math>
: <math>b = 2uv(v^2 + u^2),</math>
: <math>d = 2uv(v^2 - u^2),</math>
for coprime integers ''u'', ''v'' with {{nowrap|''v'' > ''u''}}. The geometric interpretation is that ''a'' and ''b'' are the integer legs of a right triangle and ''d'' is the integer altitude to the hypotenuse. Then the hypotenuse itself is the integer
: <math>c = (v^2 + u^2)^2,</math>
so {{nowrap|(''a'', ''b'', ''c'')}} is a [[Pythagorean triple]].

==== ''n'' < −2 ====
There are no solutions in integers for {{nowrap|1=''a''<sup>''n''</sup> + ''b''<sup>''n''</sup> = ''c''<sup>''n''</sup>}} for integers {{nowrap|''n'' < −2}}. If there were, the equation could be multiplied through by {{nowrap|''a''<sup>{{abs|''n''}}</sup>''b''<sup>{{abs|''n''}}</sup>''c''<sup>{{abs|''n''}}</sup>}} to obtain {{nowrap|1=(''bc'')<sup>{{abs|''n''}}</sup> + (''ac'')<sup>{{abs|''n''}}</sup> = (''ab'')<sup>{{abs|''n''}}</sup>}}, which is impossible by Fermat's Last Theorem.

=== abc conjecture ===
{{Main|abc conjecture#Some consequences}}

The [[abc conjecture]] roughly states that if three positive integers ''a'', ''b'' and ''c'' (hence the name) are coprime and satisfy {{nowrap|1=''a'' + ''b'' = ''c''}}, then the [[radical of an integer|radical]] ''d'' of ''abc'' is usually not much smaller than ''c''. In particular, the abc conjecture in its most standard formulation implies Fermat's last theorem for ''n'' that are sufficiently large.<ref>{{cite book |title=Algebra |last=Lang |first=Serge |authorlink=Serge Lang |publisher=Springer-Verlag New York |series=Graduate Texts in Mathematics |volume=211 |year=2002 |page=196}}</ref><ref>{{cite journal |doi=10.1155/S1073792891000144 |title=ABC implies Mordell |first=Noam |last=Elkies |authorlink=Noam Elkies |journal=[[International Mathematics Research Notices]] |volume=1991 |issue=7 |year=1991 |pages=99–109 |quote=Our proof generalizes the known implication "effective ABC [right arrow] eventual Fermat" which was the original motivation for the ABC conjecture|doi-access= <!-- not free-->}}</ref><ref name="Granville-Tucker"/> The [[Szpiro conjecture#Modified Szpiro conjecture|modified Szpiro conjecture]] is equivalent to the abc conjecture and therefore has the same implication.<ref>{{cite journal | last=Oesterlé | first=Joseph | authorlink=Joseph Oesterlé | title=Nouvelles approches du "théorème" de Fermat | url= http://www.numdam.org/item?id=SB_1987-1988__30__165_0 | series=Séminaire Bourbaki exp 694 |mr=992208 | year=1988 | journal=Astérisque | issn=0303-1179 | issue=161 | pages=165–186}}</ref><ref name="Granville-Tucker"/> An effective version of the abc conjecture, or an effective version of the modified Szpiro conjecture, implies Fermat's Last Theorem outright.<ref name="Granville-Tucker">{{cite journal | last1 = Granville | first1 = Andrew | authorlink1=Andrew Granville | last2 = Tucker | first2 = Thomas | year = 2002 | title = It's As Easy As abc | url = https://www.ams.org/notices/200210/fea-granville.pdf | journal = Notices of the AMS | volume = 49 | issue = 10| pages = 1224–1231 }}</ref>

== Prizes and incorrect proofs ==
In 1816, and again in 1850, the [[French Academy of Sciences]] offered a prize for a general proof of Fermat's Last Theorem.{{sfn|Aczel|1996|p=69|ps=}}<ref>Singh, p. 105</ref> In 1857, the academy awarded 3,000 francs and a gold medal to Kummer for his research on ideal numbers, although he had not submitted an entry for the prize.{{sfn|Aczel|1996|p=69|ps=}} Another prize was offered in 1883 by the Academy of Brussels.<ref name="Koshy_2001" >{{cite book | author = Koshy T | year = 2001 | title = Elementary number theory with applications | publisher = Academic Press | location = New York | isbn = 978-0-12-421171-1 | page = 544}}</ref>

In 1908, the German industrialist and amateur mathematician [[Paul Wolfskehl]] bequeathed 100,000 [[German gold mark|gold marks]]—a large sum at the time—to the Göttingen Academy of Sciences to offer as a prize for a complete proof of Fermat's Last Theorem.<ref>Singh, pp. 120–125, 131–133, 295–296</ref>{{sfn|Aczel|1996|p=70|ps=}} On 27 June 1908, the academy published nine rules for awarding the prize. Among other things, these rules required that the proof be published in a peer-reviewed journal; the prize would not be awarded until two years after the publication; and that no prize would be given after 13 September 2007, roughly a century after the competition was begun.<ref>Singh, pp. 120–125</ref> Wiles collected the Wolfskehl prize money, then worth $50,000, on 27 June 1997.<ref>Singh, p. 284</ref> In March 2016, Wiles was awarded the Norwegian government's [[Abel prize]] worth €600,000 for "his stunning proof of Fermat's Last Theorem by way of the modularity conjecture for semistable elliptic curves, opening a new era in number theory".<ref>{{cite web
 | url = http://www.abelprize.no/c67107/binfil/download.php?tid=67059
 | title = The Abel Prize citation 2016
 | date = March 2016
 | website = The Abel Prize
 | publisher = The Abel Prize Committee
 | access-date = 16 March 2016
 | archive-date = 20 May 2020
 | archive-url = https://web.archive.org/web/20200520195920/https://www.abelprize.no/c67107/binfil/download.php?tid=67059
 | url-status = dead
 }}</ref>

Prior to Wiles's proof, thousands of incorrect proofs were submitted to the Wolfskehl committee, amounting to roughly {{convert|10|ft|m|abbr=off|sp=us}} of correspondence.<ref>Singh, p. 295</ref> In the first year alone (1907–1908), 621 attempted proofs were submitted, although by the 1970s, the rate of submission had decreased to roughly 3–4 attempted proofs per month. According to some claims, [[Edmund Landau]] tended to use a special preprinted form for such proofs, where the location of the first mistake was left blank to be filled by one of his graduate students.<ref>''[https://books.google.com/books?id=e04FEAAAQBAJ&pg=PA16 Wheels, Life and Other Mathematical Amusements]'', Martin Gardner</ref> According to F. Schlichting, a Wolfskehl reviewer, most of the proofs were based on elementary methods taught in schools, and often submitted by "people with a technical education but a failed career".<ref>Singh, pp. 295–296</ref> In the words of mathematical historian [[Howard Eves]], "Fermat's Last Theorem has the peculiar distinction of being the mathematical problem for which the greatest number of incorrect proofs have been published."<ref name="Koshy_2001"/>

== In popular culture ==
{{Main|Fermat's Last Theorem in fiction}}

The popularity of the theorem outside science has led to it being described as achieving "that rarest of mathematical accolades: A niche role in [[Popular culture|pop culture]]."<ref>{{Cite web |last=Garmon |first=Jay |date=2006-02-21 |title=Geek Trivia: The math behind the myth |url=https://www.techrepublic.com/article/geek-trivia-the-math-behind-the-myth/ |access-date=2022-05-21 |website=TechRepublic |language=en-US }}</ref>[[File:Czech stamp 2000 m259.jpg|thumb|Czech postage stamp commemorating Wiles' proof]]

[[Arthur Porges]]' 1954 short story "[[The Devil and Simon Flagg]]" features a [[mathematician]] who bargains with the [[Devil]] that the latter cannot produce a proof of Fermat's Last Theorem within twenty-four hours.<ref>{{Cite journal|last=Kasman|first=Alex|date=January 2003|title=Mathematics in Fiction: An Interdisciplinary Course|journal=[[PRIMUS (journal)|PRIMUS]]|language=en|volume=13|issue=1|pages=1–16|doi=10.1080/10511970308984042|s2cid=122365046 |issn=1051-1970}}</ref>

In ''[[The Simpsons]]'' episode "[[The Wizard of Evergreen Terrace]]", [[Homer Simpson]] writes the equation {{nowrap|1=3987<sup>12</sup> + 4365<sup>12</sup> = 4472<sup>12</sup>}} on a blackboard, which appears to be a [[counterexample]] to Fermat's Last Theorem. The equation is wrong, but it appears to be correct if entered in a calculator with 10 [[significant figures]].<ref name=SinghSimpsons>{{cite book |url=https://books.google.com/books?id=feg_AQAAQBAJ&pg=PA35 |title=The Simpsons and Their Mathematical Secrets |last=Singh |first=Simon |authorlink=Simon Singh |pages=35–36 |year=2013 |publisher=A&C Black |isbn=978-1-4088-3530-2 |language=en}}</ref>

In the ''[[Star Trek: The Next Generation]]'' episode "[[The Royale]]", [[Jean-Luc Picard|Captain Picard]] states that the theorem is still unproven in the 24th century. The proof was released five years after the episode originally aired.<ref>{{Cite web |last=Moseman |first=Andrew |date=2017-09-01 |title=Here's a Fun Math Goof in 'Star Trek: The Next Generation' |url=https://www.popularmechanics.com/science/math/news/a28040/fermats-last-theorem-star-trek-the-next-generation/ |access-date=2023-06-09 |website=Popular Mechanics |language=en-US}}</ref>

== See also ==
{{Portal|Mathematics}}
* [[Euler's sum of powers conjecture]]
* [[Proof of impossibility]]
* [[Sums of powers]], a list of related conjectures and theorems
* [[Wall–Sun–Sun prime]]

== Footnotes ==
{{reflist|group="note"|30em}}

== References ==
{{reflist}}

== Bibliography ==
{{refbegin|30em}}
* {{cite book
 | last=Aczel | first=Amir | authorlink=Amir Aczel
 | date=1996
 | title=Fermat's Last Theorem: Unlocking the Secret of an Ancient Mathematical Problem
 | publisher=Four Walls Eight Windows
 | isbn=978-1-56858-077-7
 | url=https://archive.org/details/fermatslasttheor00acz_pep
 }}
* {{cite book
 | last1 = Dickson | first1 = LE | authorlink = Leonard Eugene Dickson
 | year = 1919
 | title = History of the Theory of Numbers
 | volume = II
 | series = Diophantine Analysis
 | publisher = Chelsea Publishing | location = New York
 | pages = 545–550, 615–621, 688–691, 731–776
 }}
* {{cite book
 | last1 = Edwards | first1 = HM | authorlink = Harold Edwards (mathematician)
 | year = 1996 | orig-year=1977
 | title = Fermat's Last Theorem. A Genetic Introduction to Algebraic Number Theory
 | series = Graduate Texts in Mathematics
 | volume = 50
 | publisher = Springer-Verlag | location = New York
 | isbn = 978-0-387-90230-2
 }}
* {{cite book
 | last = Friberg | first = Joran
 | year = 2007
 | title = Amazing Traces of a Babylonian Origin in Greek Mathematics
 | publisher = World Scientific Publishing Company
 | isbn = 978-981-270-452-8
 }}
* {{cite journal
 | last1 = Kleiner | first1 = I
 | year = 2000
 | title = From Fermat to Wiles: Fermat's Last Theorem Becomes a Theorem
 | journal = [[Elemente der Mathematik]]
 | volume = 55
 | pages = 19–37
 | url = http://math.stanford.edu/~lekheng/flt/kleiner.pdf
 | doi = 10.1007/PL00000079
 | s2cid = 53319514
 | url-status = dead
 | archive-url = https://web.archive.org/web/20110608052614/http://math.stanford.edu/~lekheng/flt/kleiner.pdf
 | archive-date = 8 June 2011
 }}
* {{cite book
 | last1 = Mordell | first1 = LJ | authorlink=Louis Mordell
 | year = 1921
 | title = Three Lectures on Fermat's Last Theorem
 | publisher = Cambridge University Press | location = Cambridge
 }}
* {{cite book
 | last1 = Manin | first1 = Yuri Ivanovic
 | last2 = Panchishkin | first2 = Alekseĭ Alekseevich
 | year = 2007
 | title = Introduction to Modern Number Theory
 | chapter = Fundamental problems, Ideas and Theories
 | series = Encyclopedia of Mathematical Sciences
 | volume = 49
 | edition = 2nd
 | publisher=Springer | location=Berlin Hedelberg
 | isbn = 978-3-540-20364-3
 }}
* {{cite book
 | last1 = Ribenboim | first1=P | authorlink=Paulo Ribenboim
 | year = 2000
 | title = Fermat's Last Theorem for Amateurs
 | publisher = Springer-Verlag | location = New York
 | isbn = 978-0-387-98508-4
 }}
* {{cite book
 | last1 = Singh | first1 = S | authorlink = Simon Singh
 | year = 1998
 | title = Fermat's Enigma
 | publisher = Anchor Books | location = New York
 | isbn = 978-0-385-49362-8
 | title-link = Fermat's Enigma
 }}
* {{cite book
 | last1=Stark | first1=H | authorlink=Harold Stark
 | year = 1978
 | title = An Introduction to Number Theory
 | publisher = MIT Press
 | isbn = 0-262-69060-8
 | url = https://archive.org/details/introductiontonu00star_0
 }}
{{refend}}

== Further reading ==
{{refbegin|30em}}
* {{cite book | last=Bell | first=Eric T. | title=The Last Problem | location=New York | isbn=978-0-88385-451-8 | date=1998 | publisher=The Mathematical Association of America | orig-year=1961 }}
* {{cite book | last=Benson | first=Donald C. | title=The Moment of Proof: Mathematical Epiphanies | publisher=Oxford University Press | isbn=978-0-19-513919-8 | date=2001}}
* {{cite book | last=Brudner | first=Harvey J. | authorlink=Harvey J. Brudner | title=Fermat and the Missing Numbers | publisher=WLC, Inc | isbn=978-0-9644785-0-3 | year=1994 }}
* {{cite journal | author = Faltings G |date=July 1995 | url = https://www.ams.org/notices/199507/faltings.pdf | title=The Proof of Fermat's Last Theorem by R. Taylor and A. Wiles | journal=[[Notices of the American Mathematical Society]] | volume=42 | issue=7 | pages=743–746 | issn=0002-9920 | authorlink=Gerd Faltings }}
* {{cite book | last=Mozzochi | first=Charles | title=The Fermat Diary | date=2000 | isbn=978-0-8218-2670-6 | publisher=American Mathematical Society }}
* {{cite book | author = Ribenboim P | year = 1979 | title = 13 Lectures on Fermat's Last Theorem | publisher = Springer Verlag | location = New York | isbn = 978-0-387-90432-0| authorlink=Paulo Ribenboim }}
* {{cite journal | last = Saikia | first = Manjil P | date = July 2011 | url = http://www.manjilsaikia.in/publ/projects/kummerFLT.pdf | title = A Study of Kummer's Proof of Fermat's Last Theorem for Regular Primes | journal = IISER Mohali (India) Summer Project Report | bibcode = 2013arXiv1307.3459S | arxiv = 1307.3459 | access-date = 9 March 2014 | archive-url = https://web.archive.org/web/20150922030715/http://www.manjilsaikia.in/publ/projects/kummerFLT.pdf | archive-date = 22 September 2015 | url-status = dead }}
* {{cite book | last=Stevens | first=Glenn | authorlink=Glenn H. Stevens | chapter=An Overview of the Proof of Fermat's Last Theorem | title=Modular Forms and Fermat's Last Theorem | location=New York | publisher=Springer | year=1997 | isbn=0-387-94609-8 | pages=1–16 | chapter-url=https://books.google.com/books?id=Va-quzVwtMsC&pg=PA1 }}
* {{cite book | last=van der Poorten | first=Alf | title=Notes on Fermat's Last Theorem | date=1996 | publisher=WileyBlackwell | isbn=978-0-471-06261-5 | url-access=registration | url=https://archive.org/details/notesonfermatsla0000vand }}
{{refend}}

== External links ==
{{Wikibooks}}
{{Wikiquote}}
{{Commons category}}
{{refbegin}}
* {{cite web | last1=Daney | first1=Charles | year=2003 | url=http://cgd.best.vwh.net/home/flt/flt01.htm | title=The Mathematics of Fermat's Last Theorem | access-date=5 August 2004 | url-status=dead | archive-url=https://web.archive.org/web/20040803221632/http://cgd.best.vwh.net/home/flt/flt01.htm | archive-date=3 August 2004 }}
* {{cite web | last1=Elkies | first1=Noam D. | url=http://www.math.harvard.edu/~elkies/ferm.html | title=Tables of Fermat "near-misses" – approximate solutions of ''x''<sup>''n''</sup> + ''y''<sup>''n''</sup> = ''z''<sup>''n''</sup> }}
* {{cite web | last1=Freeman | first1=Larry | year=2005 |url=http://www.fermatslasttheorem.blogspot.com | title=Fermat's Last Theorem Blog}} Blog that covers the history of Fermat's Last Theorem from Fermat to Wiles.
* {{SpringerEOM | title=Fermat's last theorem | id=p/f110070|mode=cs1 }}
* {{cite journal
 | last1 = Ribet | first1 = Kenneth A.
 | year = 1995
 | arxiv = math/9503219
 | doi = 10.1090/S0273-0979-1995-00616-6
 | issue = 4
 | journal = Bulletin of the American Mathematical Society
 | mr = 1322785
 | pages = 375–402
 | series = New Series
 | title = Galois representations and modular forms
 | volume = 32
 | s2cid = 16786407
}} Discusses various material that is related to the proof of Fermat's Last Theorem: elliptic curves, modular forms, Galois representations and their deformations, Frey's construction, and the conjectures of Serre and of Taniyama–Shimura.
* {{cite web|author=Shay, David|year=2003| url=http://shayfam.freevar.com/david/flt/index.htm| title=Fermat's Last Theorem| access-date=14 January 2017}} The story, the history and the mystery.
* {{MathWorld | urlname=FermatsLastTheorem | title=Fermat's Last Theorem}}
* {{cite web | vauthors = O'Connor JJ, Robertson EF | year = 1996 | url = http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Fermat's_last_theorem.html | title = Fermat's last theorem | access-date = 5 August 2004 | url-status = dead | archive-url = https://web.archive.org/web/20040804045854/http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Fermat%27s_last_theorem.html | archive-date = 4 August 2004 }}
* {{cite web | url=https://www.pbs.org/wgbh/nova/proof/ | title=The Proof | publisher=[[PBS]]}} The title of one edition of the PBS television series NOVA, discusses Andrew Wiles's effort to prove Fermat's Last Theorem.
* {{cite web | url=http://vimeo.com/18216532 | title= Documentary Movie on Fermat's Last Theorem (1996) }} Simon Singh and John Lynch's film tells the story of Andrew Wiles.
{{refend}}

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